use the second derivative test to classify the relative extrema if the test applies
Answers
Answer
The answer is:
[tex](x,f(x))=(0,256)[/tex]SOLUTION
Problem Statement
The question gives us a polynomial expression and we are asked to find the relative maxima using the second derivative test.
The function given is:
[tex](3x^2+16)^2[/tex]Method
To find the relative maxima, there are some steps to perform.
1. Find the first derivative of the function
2. Equate the first derivative to zero and solve for x.
3. Find the second derivative of the function.
4. Apply the second derivative test:
This test says:
[tex]\begin{gathered} \text{ If }a\text{ is one of the roots of the equation from the first derivative, then,} \\ f^{\doubleprime}(a)>0\to\text{There is a relative minimum} \\ f^{\doubleprime}(a)<0\to\text{There is a relative maximum} \end{gathered}[/tex]5. Find the Relative Minimum
Implementation
1. Find the first derivative of the function
[tex]\begin{gathered} f(x)=(3x^2+16)^2 \\ \text{Taking the first derivative of both sides, we have:} \\ f^{\prime}(x)=6x\times2(3x^2+16) \\ f^{\prime}(x)=12x(3x^2+16) \end{gathered}[/tex]2. Equate the first derivative to zero and solve for x.
[tex]\begin{gathered} f^{\prime}(x)=12x(3x^2+16)=0 \\ \text{This implies that,} \\ 12x=0\text{ OR }3x^2+16=0 \\ \therefore x=0\text{ ONLY} \\ \\ \text{Because }3x^2+16=0\text{ has NO REAL Solutions} \end{gathered}[/tex]This implies that there is ONLY ONE turning point/stationary point at x = 0
3. Find the second derivative of the function:
[tex]\begin{gathered} f^{\prime}(x)=12x(3x^2+16) \\ f^{\doubleprime}(x)=12(3x^2+16)+12x(6x) \\ f^{\doubleprime}(x)=36x^2+192+72x^2 \\ \therefore f^{\doubleprime}(x)=108x^2+192 \end{gathered}[/tex]4. Apply the second derivative test:
[tex]\begin{gathered} f^{\doubleprime}(x)=108x^2+192 \\ a=0,\text{ which is the root of the first derivative }f^{\prime}(x) \\ f^{\doubleprime}(a)=f^{\doubleprime}(0)=108(0)^2+192 \\ f^{\doubleprime}(0)=192>0 \\ \\ By\text{ the second derivative test,} \\ f^{\doubleprime}(0)>0,\text{ thus, there exists a relative minimum at }x=0\text{ } \\ \\ \text{ Thus, we can find the relative minimum when we substitute }x=0\text{ into the function }f(x) \end{gathered}[/tex]5. Find the Relative Minimum:
[tex]\begin{gathered} f(x)=(3x^2+16)^2 \\ \text{substitute }x=0\text{ into the function} \\ f(0)=(3(0)^2+16)^2 \\ f(0)=16^2=256 \\ \\ \text{Thus, the minimum value of the function }f(x)\text{ is }256 \\ \\ \text{The coordinate for the relative minimum for the function }(3x^2+16)^2\text{ is:} \\ \mleft(x,f\mleft(x\mright)\mright)=\mleft(0,f\mleft(0\mright)\mright) \\ \text{But }f(0)=256 \\ \\ \therefore(x,f(x))=(0,256) \end{gathered}[/tex]Since the function has ONLY ONE turning point, and the turning point is a minimum value, then THERE EXISTS NO MAXIMUM VALUE
Final Answer
The answer is:
[tex](x,f(x))=(0,256)[/tex]
Related Questions
The path of the baseball follows the equation h= -4.9t^2 + 60t + 1.5 where h represents the height of the baseball, t seconds after the baseball was hit. How long will it take the baseball to return to the ground?
Answers
SOLUTION
Given the question in the question tab, the following are the steps to solve the problem:
Step 1: Write out the equation for the path of the baseball where h is height and t is time in seconds
[tex]h=-4.9t^2+60t+1.5[/tex]Step 2: Rewrite the new equation
The height of the baseball when it returns to the ground is zero(0). Therefore, at that point where the baseball returns to the ground, the function becomes:
[tex]0=-4.9t^2+60t+1.5[/tex]Step 3: We solve the quadratic equation to get the value of t:
[tex]\begin{gathered} 0=-4.9t^2+60t+1.5 \\ u\sin g\text{ quadratic formula which states that:} \\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ a=-4.9,b=60,c=1.5 \\ \text{Substituting the values, we have:} \\ \frac{-60\pm\sqrt[]{60^2-4(-4.9)(1.5)}}{2(-4.9)} \\ =\frac{-60\pm\sqrt[]{3600+29.4}}{-9.8} \\ =\frac{-60\pm60.2445}{-9.8} \\ =\frac{-60+60.2445}{-9.8}\text{ or }\frac{-60-_{}60.2445}{-9.8} \\ =\frac{0.2445}{-9.8}or\frac{-120.2445}{-9.8} \\ t=-0.024948979\text{ or }12.26984184 \\ t\approx-0.0249\text{ or 12.270} \end{gathered}[/tex]Since the value for time cannot be negative, hence the time it will it take the baseball to return to the ground is approximately 12.270 seconds
help meeeeeeeeee pleaseee !!!!!
Answers
The value of the composite function is as follows:
(gof)(5) = 6How to find composite function?The composite function can be solved as follows:
Composite functions are when the output of one function is used as the input of another.
In other words, a composite function is a function that depends on another function.
f(x) = x² - 6x + 2
g(x) = -2x
Therefore,
(gof)(5) = g(f(5))
So we need to find g(f(x)) first.
Therefore,
g(f(x)) = -2(x² - 6x + 2)
g(f(x)) = - 2x² + 12x - 4
Therefore,
g(f(x)) = - 2x² + 12x - 4
(gof)(5) = g(f(5)) = - 2(5)² + 12(5) - 4
(gof)(5) = g(f(5)) = -50 + 60 - 4
(gof)(5) = g(f(5)) = 6
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Omoro bought 2 2/3 pounds of takis that he is going to bring to school for lunch each day in plastic bags that carry 1/8 of a pound.how many bags can omoro fill completely ?
Answers
Given:
Amount of takis bought = 2⅔ pounds
Amount the plastic bag can carry = 1/8 pounds
First convert 2⅔ to a simple fraction:
2⅔ = 8/3
To find the amount of bags Omoro can fill completely, we have to divide the amount of takis bought by the amount of takis the plastic bag can carry:
(8/3) ÷ (1/8)
[tex]\begin{gathered} =\text{ }\frac{8}{3}\text{ }\ast\text{ }\frac{8}{1} \\ =\text{ }21.3\text{ bags} \end{gathered}[/tex]Therefore, Omoro can fill approximately 21 bags completely
ANSWER:
21 bags
Graph the intersection or union, as appropriate, of the solutions of the pair of linear inequalities
Answers
See graph below
Expanation:The given inequalities:
[tex]\begin{gathered} x\text{ + y }\leq\text{ 4} \\ x\text{ }\ge2 \end{gathered}[/tex]To plot the graphs, we will assing values to x in order to get the corresponding values of y for each of the inequality:
let x = 0, 2, 4
x + y = 4
from the above: y = 4 - x
when x = 0
y = 4
when x = 2
y = 4 - 2 = 2
when x = 4
y = 4 -4 = 0
we only have x in the second inequality
we will have a vertical line for x = 2
But the shading will be towards the right because the inequality is greater than x
plotting the graph:
The solution of the inequalities is the point of intersection of both graphs (the darker shade)
Find the probability that a dart hits one of the shaded areas. Thewhite figure is a rectangle. Be sure to show all work.
Answers
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Get the angles of the hexagon
The internal angles of an hexagon is given as:
[tex]\begin{gathered} \frac{180(n-2)}{n} \\ n=6\text{ since hexagon has 6 sides} \\ So\text{ we have:} \\ \frac{180(6-2)}{6}=\frac{180(4)}{6}=\frac{720}{6}=120\degree \end{gathered}[/tex]Therefore each angle of the hexagon is 120 degrees.
STEP 2: find the length of the sides
We remove the right triangles as seen below:
Using the special right triangles, we have:
STEP 3: find the area of the extracted triangle above
[tex]\begin{gathered} b=1,h=\sqrt{3} \\ Area=\frac{1}{2}\cdot1\cdot\sqrt{3}=\frac{\sqrt{3}}{2}units^2 \end{gathered}[/tex]Since there are two right triangles, we multiply the area by 2 to have:
[tex]Area=2\cdot\frac{\sqrt{3}}{2}=\sqrt{3}[/tex]There are two triangles(both sides), therefore the total area of the shaded area will be:
[tex]\sqrt{3}\cdot2=2\sqrt{3}[/tex]STEP 4: Find the area of the whole hexagon
[tex]\begin{gathered} Area=\frac{3\sqrt{3}s^2}{2} \\ s=hypotenuse\text{ of the right triangle}=2 \\ Area=\frac{3\sqrt{3}\cdot4}{2}=6\sqrt{3} \end{gathered}[/tex]STEP 5: Find the probability
[tex]\begin{gathered} Probability=\frac{possible\text{ area}}{Total\text{ area}} \\ \\ Possible\text{ area}=2\sqrt{3} \\ Total\text{ area}=6\sqrt{3} \\ \\ Probability=\frac{2\sqrt{3}}{6\sqrt{3}}=\frac{1}{3}=0.3333 \end{gathered}[/tex]Hence, the probability that the dart hits one of the shaded areas is approximately 0.3333
Solve the system of inequalities by graphing.y\ge-3
Answers
Represent each sum as a single rational number. -14+(-8/9) due tomorrow pls answer
Answers
the given expression is
-14 + (-8/9)
so,
[tex]\begin{gathered} =-14+\frac{-8}{9} \\ =-14-\frac{8}{9} \end{gathered}[/tex][tex]\begin{gathered} =\frac{-126-8}{9} \\ =-\frac{134}{9} \end{gathered}[/tex][tex]=-\frac{134}{9}=-14\frac{8}{9}[/tex]so the answer is -14 8/9 or -134/9
Which best represents the number of square centimeters in a square foot?A 366 square centimeters B 144 square centimeters C 930 square centimeters D 61 square centimeters
Answers
Answer:
C. 930 square centimeters
Explanation:
First, recall the standard conversion between cm and ft.
[tex]1\text{ ft}=30.48\operatorname{cm}[/tex]Therefore:
[tex]\begin{gathered} (1\times1)ft^2=(30.48\times30.48)cm^2 \\ =929.03\operatorname{cm}^2 \\ \approx930\text{ square cm} \end{gathered}[/tex]The correct choice is C.
Can you write on the paper/photo? So can write on my paper too and write it down
Answers
Answer:
1) 4x + 12
2) new area = 16x + 48
3) Yes, the ratio is the same for positive values of x
Explanation:
The distributive property of multiplication is shown below
a(b + c) = ab + ac
The area of the given rectangle is expressed as
Area = 4(x + 3)
By applying the distributive property, it becomes
4 * x + 4 * 3
= 4x + 12
The equivalent expression is
4x + 12
If the dimensions of the rectangle are doubled, then
new length = 2(x + 3) = 2x + 6
new width = 4 * 2 = 8
Thus,
new area = 8(2x + 6) = 8 * 2x + 8 * 6
new area = 16x + 48
We would input values of x into both areas and find their ratios
For x = 1,
area = 4(1) + 12 = 16
new area = 16(1) + 48 = 64
ratio = 16/64 = 1/4
For x = 2,
area = 4(2) + 12 = 20
new area = 16(2) + 48 = 80
ratio = 20/80 = 1/4
For x = 3,
area = 4(3) + 12 = 24
new area = 16(3) + 48 = 96
ratio = 24/96 = 1/4
Thus, the ratio is the same for positive values of x
In an all boys school, the heights of the student body are normally distributed with a mean of 70 inches and a standard deviation of 3 inches. What is the probability that a randomly selected student will be taller than 71 inches tall, to the nearest thousandth?
Answers
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Answer:
0.010
Step-by-step explanation:
We solve the above question using z score formula
z = (x-μ)/σ, where
x is the raw score = 63 inches
μ is the population mean = 70 inches
σ is the population standard deviation = 3 inches
For x shorter than 63 inches = x < 63
Z score = x - μ/σ
= 63 - 70/3
= -2.33333
Probability value from Z-Table:
P(x<63) = 0.0098153
Approximately to the nearest thousandth = 0.010
Therefore, the probability that a randomly selected student will be shorter than 63 inches tall, to the nearest thousandth is 0.010.
The table displays the mean name length for seven samples of students.Sample1Mean Name Length5.47.1236.345.2566.04.976.2What can be said about the variation between the sample means?The variation between the sample means is small.The variation between the sample means is large.The variation shows that the values are far apart.The variation cannot be used to make predictions.
Answers
First option is correct.
For all the sample sizes, the sample mean is close to 6, give or take (
Solve the equation below by completing the square and then solving for x.2 + 14x + 24 = 0A. X=-12 or x= -2B. x=2 or x=-2C. X=-2 or x=-5D. X= 8 or x= 3
Answers
2x^2 + 14x + 24 = 0
x^2 + 7x + 12 = 0
Then
x^ + (14/2) x + 12 + [7x + 49 ] = 7x + 49
Now form square
[ x + 7x + 7x + 49 ] = 7x + 49 - 12
[ x + 7]^2 = 7x + 37
Then now find x
x^2 + 7x = -6
x^2 + 7x + 7^2/4 = 7^2/4 - 6
then
( x + 7/2)^2 = 49/4 - 24/4
(x + 7/2) = ±√ (25/4)
. x = 5/2 - 7/2
and. x = 5/2 + 7/2
Then solutions are
x = -2
x= -12
ANSWER IS
OPTION A
Find the area of the shapes below. Must show all steps includingformula and units! If needed, round your answer to the nearest tenth. This is a parallelogram
Answers
Answer: Area = 120 cm^2
Explanation:
The formula for calculating the area of a parallelogram is expressed as
Area = base x height
From the information given,
base = 15
height = 8
Area = 15 x 8
Area = 120 cm^2
Fill in the table using this function rule. y = -10x +3 y X 6 ? 1 0 a 1
Answers
the function is
[tex]y=-10x+3[/tex]we must replace the value of x and obtain y
x=-5
[tex]\begin{gathered} y=-10(-5)+3 \\ y=50+3 \\ y=53 \end{gathered}[/tex]x=-1
[tex]\begin{gathered} y=-10(-1)+3 \\ y=13 \end{gathered}[/tex]x=0
[tex]\begin{gathered} y=-10(0)+3 \\ y=3 \end{gathered}[/tex]x=1
[tex]\begin{gathered} y=-10(1)+3 \\ y=-7 \end{gathered}[/tex]Find the indicated quantity, given u = (4, -9), v = (-4, -7).Step 4 of 4: Find (u • v)4v.
Answers
Answer:
[tex]\begin{equation*} \langle-1316,-2303\operatorname{\rangle} \end{equation*}[/tex]Explanation:
Given the vectors:
[tex]\begin{gathered} u=\langle4,-9\rangle \\ v=\langle-4,-7\rangle \end{gathered}[/tex]The dot product of u and v is calculated below:
[tex]\begin{gathered} u\cdot v=4\times-4+-9\times-7 \\ =-16+63 \\ =47 \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} (u\cdot v)4v=47\times4\langle-4,-7\rangle \\ =329\langle-4,-7\rangle \\ =\langle-4\times329,-7\times329\rangle \\ =\langle-1316,-2303\operatorname{\rangle} \end{gathered}[/tex]The indicated quantity is:
[tex]\begin{equation*} \langle-1316,-2303\operatorname{\rangle} \end{equation*}[/tex]
Harold Hill borrowed $16,400 to pay for his child's education at Riverside Community College. Harold must repay the loan at the end of 15 months in one payment with 3 3/4 % of interest.
A. How much interest must Harold pay? (Round answer to the nearest cent.)
B. What is the maturity value? (Round answer to the nearest cent.)
Answers
The interest that Harold pay is $768.75 and his maturity value is $17168.75.
Harold Hill borrowed $16,400
Harold must repay the loan at the end of 15 months in one payment with 3 3/4 % of interest
First we need to calculate the interest amount
= loan amount x rate of interest x number of months
interest = (16400 x 3 3/4 x 15/12)/100
interest = $768.75
The maturity value = loan amount + interest
= 16400 + 768.75
= 17168.75
Therefore, the interest that Harold pay is $768.75 and his maturity value is $17168.75.
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I will send a picture of the problem and or question
Answers
The equivalency for grams to centigrams is:
1 gram = 100centigrams
To convert the units you can apply cross multiplication:
1gr_____100cgr
443gr____xcgr
[tex]\begin{gathered} \frac{100}{1}=\frac{x}{443} \\ x=443\cdot100=44300 \end{gathered}[/tex]This means that 443 grams equals to 44300 centigrams
*-*-*-*
The scale is done in a base of 10 and the grams are in its center with value 1.
To convert from smaller units to grater units you have to divide the given measurement by 10
And to convert from greater units to smaller units you have to multiply by 10.
For example if you have 1mg and want to convert it to grams you have to divide the value 3 times by 10, i.e. divide the value by 1000
[tex]\frac{1mg}{1000}=0.001g[/tex]If you want to convert 1 Kg into 1 decagram, multiply the value two times by 10, i.e. multiply it by 100
[tex]1\operatorname{kg}\cdot100=100\text{dag}[/tex]A car travels 273 miles in 6 hours. How muchtime will it take traveling 378 miles
Answers
hello
the car travels 273 miles in 6 hours, how many hours will it take to travel 378 miles.
let the number of unknown hours be represented by x
[tex]\begin{gathered} 273mi=6\text{hrs} \\ 378mi=\text{xhr} \\ \text{cross multiply bith sides} \\ 273\times x=6\times378 \\ 273x=2268 \\ \text{divide both sides by 273} \\ \frac{273x}{273}=\frac{2268}{273} \\ x=8.3076\text{hrs} \end{gathered}[/tex]the car spent approximately 8.31 hours to travel a distance of 378 miles
Please provide deep explanation, so i can understand and learn. Thank you
Answers
Assume the height of the box is x.
5 reams of paper have 5 x 500 = 2500 sheets of paper.
This means that each sheet of paper has a thickness of x/2500.
Two sheets of paper have a thickness of 2 times x/2500.
Simplifying the fraction:
[tex]2\cdot\frac{x}{2500}=\frac{x}{1250}[/tex]Two sheets of paper have a thickness of 1/1250th of the height of the box.
Assume the height is x = 20 cm, then two sheets are 20/1250 = 0.016 cm thick.
Linda's medicine bottlesays "If you will be driving, then youshould not take this medicine." What arethe inverse, converse, and thecontrapositive of this statement?
Answers
For two statements p and q, and the compounded statement "If p, then q", we have the following definitions for the inverse, converse, and contrapositive of this compounded statement:
inverse: If not p, then not q.
converse: If q, then p.
contrapositive: If not q, then not p.
So, for the presented statement, i. e., "If you will be driving, then you should not take this medicine" we have:
p: you will be driving
q: you should not take this medicine
Notice that:
not p: you will not be driving
not q: you may take this medicine
Then, using the above definitions, we write:
inverse: If you will not be driving, then you may take this medicine.
converse: If you should not take this medicine, then you will drive.
contrapositive: If you may take this medicine, then you will not be driving.
Margo borrows $1200, agreeing to pay it back with 4% annual interest after 17 months. How much interest will she pay?
Answers
Answer:
$68
Step-by-step explanation:
P = $1200
R = 4%
T = 17months (Convert to years; 17 months ÷ 12 months)
Formular for Interest; I = PRT
100
I = $1200 × 4 × 17
100 × 12
I = $68
Identify the property of real numbers illustrated in the following equation.(+6) + [y? • (-4)] = [y2 • (-4)] + (-6)
Answers
Given
[tex]\mleft(+6\mright)+\mleft[y^2•(-4)\mright]=\mleft[y^2•(-4)\mright]+(-6)[/tex]Answer
Commutative property of addition
Two question I want to verify my answerSolve for y in terms of x 2x =1-5yAnd Simplify the given expression Write answer with a positive exponent (X^-3/y^4)^-4
Answers
Part 1
we have
2x =1-5y
solve for y
step 1
Adds 5y both sides
2x+5y=1
step 2
subtract 2x both sides
5y=-2x+1
step 3
Divide by 5 on both sides
y=-(2/5)x+1/5
Part 2
we have the expression
[tex](\frac{x^{-3}}{y^4})^{-6}=(\frac{y^4}{x^{-3}})^6=(y^4x^3)^6=y^{(24)}x^{(18)}[/tex]Write an equation in slope-intercept form for the line through (-1, 1) and (0,3).
Answers
The slope intercept form of a line can be written as:
[tex]y=mx+b[/tex]where m is the slope and b is the y-intercept.
We have two points of the line: (-1,1) and (0,3).
Knowing that for x=0, the value of y=3 tells us that the y-intercept b is b=3:
[tex]\begin{gathered} y=mx+b \\ 3=m\cdot0+b \\ 3=b \end{gathered}[/tex]Using the other point and replacing the values of x and y in the equation we can calculate the value of the slope m:
[tex]\begin{gathered} y=mx+3 \\ 1=m\cdot(-1)+3 \\ 1-3=-m \\ -2=-m \\ m=2 \end{gathered}[/tex]Then, with m=2 and b=3, the equation becomes:
[tex]y=2x+3[/tex]Answer: y=2x+3
1. Canada has the longest coastline of any country. It is 202,080 km. China has 22,147 km of borders - more than any other countries. What is the difference between the two lengths? Label your answer!!
Answers
Canada has a coastline if 202,080km and China has 22,147 of borders, so the difference between them can be writen like this:
[tex]202080-22147[/tex]And we can made this operation so the answer is:
[tex]202080-22147=179,933[/tex]This means that the coastline of canada is 179,933 longer than the border of china
I need help with this problem Math relatedsimplify in radical form:^3{120a^4b^5c
Answers
Sara, this is the solution:
∛ 120a^4b^5c
Lets solve factor by factor:
∛ 120 = ∛ 8 * 15 = 2∛ 15
∛a^4 = a^4/3
∛b^5 = b^5/3
∛c (We can't simplify this factor)
In consequence, we have:
2a^(4/3) b^(5/3)∛ 15c
Convert the following percent to a simplified fraction: 6%
Answers
Given:
6%
To convert the given percent into simplified fraction, we first divide it by 100 as shown below:
[tex]\begin{gathered} 6\%=\frac{6}{100} \\ Simplify \\ =\frac{3}{50} \end{gathered}[/tex]Therefore, the answer is:
[tex]\frac{3}{50}[/tex]which is the BEST first step in order to solve this equation15 + 2/3 a = -5a.subtract 15 from both sides b.subtract 2/3 feom both sides c.add 5 to both sides d.multiply by 3 on both sides
Answers
In order to solve this equation, we need to isolate the variable a in one side of the equation.
Since we have the number 15 in the same side of the variable, the best first step would be removing this number 15 from this side, and we do this by subtracting 15 from both sides.
Therefore the answer is a.
Which can be the first step in finding the equation of the line that passes through the points (5,-4) and (-1,8) in slope-intercept form?8-(-4) 12-12--2Calculate -1-5Calculate 8-(-4) 12-1-5 -6Find that the point at which the line intersects with the line y = 0 is (3,0).Find that the point at which the line intersects with the line X=Y is (2, 2).
Answers
The first step to finding the equation of the line in the slope-intercept form is to find the slope.
So, to find the slope we can use the following equation:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Where (x1, y1) and (x2, y2) are points of the line.
Therefore, if we replace (x1, y1) by (5, -4) and (x2, y2) by (-1, 8), we get that the first step is to calculated:
y2 - y1 = 8 - (-4) = 8 + 4 = 12
x2 - x1 = -1 - 5 = -6
Answer: Calculate 8 - ( - 4 ) = 12
Calculate - 1 - 5 = -6
Help first one to get this correct will be marked
Answers
Answer:
1st one
Step-by-step explanation:
1st one
You have one input (x) with more than one output (y)
(-9, -9) and (-9, 6)