A curve has equation y=4x^3 -3x+3. Find the coordinates of the two stationary points. Determine whether each of the stationary points is a maximum or a minimum.

Answer:

There are two stationary points

Note that 1/2 = 0.5

==========================================================

How to get those answers:

Stationary points occur when the derivative is zero, when the graph is neither increasing nor decreasing (i.e. it's staying still at that snapshot in time).

Apply the derivative to get:

f(x) = 4x^3 - 3x + 3

f ' (x) = 12x^2 - 3

Then set it equal to zero and solve for x.

f ' (x) = 0

12x^2 - 3 = 0

12x^2 = 3

x^2 = 3/12

x^2 = 1/4

x = sqrt(1/4) or x = -sqrt(1/4)

x = 1/2 or x = -1/2

x = 0.5 or x = -0.5

----------------------

Let's do the first derivative test to help determine if we have local mins, local maxes, or neither.

Set up a sign chart as shown below. Note the three distinct regions A,B,C

The values -0.5 and 0.5 are not in any of the three regions. They are the boundaries.

The reason why I split things into regions like this is to test each region individually. We'll plug in a representative x value into the f ' (x) function.

To start off, we'll check region A. Let's try something like x = -2

f ' (x) = 12x^2 - 3

f ' (-2) = 12(-2)^2 - 3

f ' (-2) = 45

The actual result doesn't matter. All we care about is whether if its positive or negative. In this case, f ' (x) > 0 when we're in region A. This tells us f(x) is increasing on the interval [tex]-\infty < x < -0.5[/tex]

Let's check region B. I'll try x = 0.

f ' (x) = 12x^2 - 3

f ' (0) = 12(0) - 3

f ' (0) = -3

The result is negative, so f ' (x) < 0 when [tex]-0.5 < x < 0.5[/tex]. The f(x) curve is decreasing on this interval.

The change from increasing to decreasing as we pass through x = -0.5 indicates that we have a local max here.

Plug x = -0.5 into the original function to find its paired y coordinate.

f(x) = 4x^3 - 3x + 3

f(-0.5) = 4(-0.5)^3 - 3(-0.5) + 3

f(-0.5) = 4

The point (-0.5, 4) is a stationary point. More specifically, it's a local max.

Side note: This is the same as the point (-1/2, 4) when written in fraction form.

----------------------

Let's check region C

I'll try x = 2

f ' (x) = 12x^2 - 3

f ' (2) = 12(2)^2 - 3

f ' (2) = 45

The positive outcome tells us that any number from region C does the same, and f ' (x) > 0 when [tex]0.5 < x < \infty[/tex]. The function f(x) is increasing on this interval.

Region B decreases while C increases. The change from decreasing to increasing indicates we have a local min when x = 0.5

Plug this x value into the original equation

f(x) = 4x^3 - 3x + 3

f(0.5) = 4(0.5)^3 - 3(0.5) + 3

f(0.5) = 2

The local min is located at (0.5, 2) which is the other stationary point.

The graph and sign chart are shown below.

The local min is located at (0.5, 2) which is the other stationary point.

Stationary points occur when the derivative is zero, when the graph is neither increasing nor decreasing.

Apply the derivative to get:

f(x) = 4x^3 - 3x + 3

f ' (x) = 12x^2 - 3

Then set it equal to zero and solve for x.

f ' (x) = 0

12x^2 - 3 = 0

12x^2 = 3

x^2 = 3/12

x^2 = 1/4

x = 1/2 or x = -1/2

x = 0.5 or x = -0.5

The first derivative test to help determine if we have local mins, local maxes, or neither.

A = numbers to the left of -0.5

B = numbers between -0.5 and 0.5 excluding both endpoints

C = numbers to the right of 0.5

Thus, The values -0.5 and 0.5 are not in any of the three regions. They are the boundaries.

Let's try something like x = -2

f ' (x) = 12x^2 - 3

f ' (-2) = 12(-2)^2 - 3

f ' (-2) = 45

In this case, f ' (x) > 0 when we're in region A.

Let's check region B x = 0.

f ' (x) = 12x^2 - 3

f ' (0) = 12(0) - 3

f ' (0) = -3

The result is negative, so f ' (x) < 0 when f(x) curve is decreasing on this interval.

The change from increasing to decreasing as we pass through x = -0.5 indicates that we have a local max here.

Substitute x = -0.5 into the original function to find its paired y coordinate.

f(x) = 4x^3 - 3x + 3

f(-0.5) = 4(-0.5)^3 - 3(-0.5) + 3

f(-0.5) = 4

The point (-0.5, 4) is a stationary point, it's a local max.

Let's check region C x = 2

f ' (x) = 12x^2 - 3

f ' (2) = 12(2)^2 - 3

f ' (2) = 45

The positive outcome tells us that any number from region C does the same, and f ' (x) > 0 when The function f(x) is increasing on this interval.

Susbtitute this x value into the original equation

f(x) = 4x^3 - 3x + 3

f(0.5) = 4(0.5)^3 - 3(0.5) + 3

f(0.5) = 2

Hence, The local min is located at (0.5, 2) which is the other stationary point.

Learn more about equations here;

https://brainly.com/question/10413253

#SPJ2

6:4 is equivalentto blank:24

Solve the inequality and graph the solution.
j/4>2

Help me with these worksheets pls

Anything would help BUT PLS ANSWER IT CORRECTLY, If you don't know, just IGNORE THIS AND DON'T JUST COME FOR THE POINTS.

help pls and thank you​

What is the equation of this graphed line?



Enter your answer in slope-intercept form in the box.

Pls help my child has no clue and this math is too hard for me to understand!!

The sum of the first 15 terms of a geometric sequence with 7 as the first term and a common ratio of -3 is __________.

A fair coin is tossed (heads or tails) 9 times. What is the probability that the coin will land “heads” exactly 3 out of the 9 times?

Select the best estimate for the quotient.75.1÷9=

in 2011 i sold 105276 tickets.
This is 7% less then the number of tickets i sold in 2018.

How many tickets i sell in 2018?

And

Show a check of your answer.

Consider the sequence: -7, -2, 3, 8…. What is f(7)?

florida sales tax is 7%. How much money comes out of your pocket if u buy something for $100

Please help!! Thank you!!!!

What is log Subscript 5 Baseline (4 times 7) + log Subscript 5 Baseline 2 written as a single log?

3a+5=20 steps on paper

Please help meee! I will give brainlist to quickest answer !!!

What is the equation of this graphed line?

A researcher at the Centers for Disease Control and Prevention is studying the growth of a certain bacteria. He starts his experiment with 200 bacteria that grow at a rate of 2% per hour. He will check on the bacteria every 7 hours. How many bacteria will he find in 7 hours? Round your answer to the nearest whole number.

The diameter of a circle fits around the circle ______ times.

Help fast !!! I have like 5 minutes left

After the elves had hung 25% of the ornaments on the North Pole tree, there
were 480 more ornaments to hang. How many ornaments were in the box to
begin with?

(GIVING BRAINLIEST AND EXTRA POINTS) A multiplication expression with a product of 108,064 is shown below. What is the missing digit?



2
3
4
5

PLEASE HELP I WILL MARK YOU BRAINLEST!!!
please read the question on the photo and these are the choices bellow!!

A-translation, reflection
B- rotation, reflection
C-reflection, rotation
D-rotation, translation

Bob purchased 2 hot dogs and 1 hamburger for S10. Later that day, Bob had to purchase more food and
Bot 1 hot dog and 3 hamburgers for S15. What were the prices of the hamburger and hot dogs?

How can areas of composite 2-D figures be found using simple 2-D shapes?

7 x [(7 + 7) = 7
Help plz

The roots of the function f(x) = x2 – 2x – 3 are shown. What is the missing number?

x = –1 and x =

is -74.58 an integer

a. Morning rain showers dropped 1/3 of an inch of rain. Later that day, another storm passed through and dropped 1/8 inch of rain. How much more rain fell in the morning than in the afternoon?

Marcus sleeps 60 hours a week this is 5 times as many hours as he plays chess. How many hours a week does Marcus play chess?

11)
Given the function, f(x) = x2 + 5x - 3 evaluate f(-2).