Question

Find the critical numbers and the open intervals on which the function is increasing or decreasing. Use a graphing utility to verity your results.
y = x3 − 3x + 8
STEP 1:
Begin by finding the derivative of y.
y' =

Incorrect: Your answer is incorrect.
STEP 2:
Find the critical numbers. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
x =

STEP 3:
Test the intervals to determine the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.)
increasing

decreasing

Answer 1

Answer:

Step-by-step explanation:

What we are doing corresponds with the AP Calculus AB/BC Unit 5

We are finding properties of functions using the analytical properties of the derivative.

This is often called the Candidates Test(or First Derivative Test)

This is how it works.

Since we are looking for critical numbers and intervals of icnreasin

g,decreasing,etc.

we use the First Derivative Test

1. Find the derivative of the f(x),  

Here our function is

$y=x^3-3x+8\\Using\ the\ differnece\ rule,\\ and \ power \ rule \\\frac{dy}{dx} =3x^{2} -3$

2. Set the derivative equal to 0 to find the critical numbers

$3x^2-3=0\\3x^2=3\\x^2=1\\x=1 , x=-1$

(-1,1)

3. Use a sign analysis to find intervals.

Draw a number line. Plot the points (-1) and 1 on the line. This creates three intervals.

(-∞,-1)

(-1,1)

(1,∞)

For each interval, pick a number in the interval and plug it in derivative function

Let's do it.

Lest Test x=-2, 0, and 2

For x=-2 $\frac{dy}{dx}=3x^2-3\\ \frac{dy}{dx}(-2)=9$

For x=0, dy/dx{0)=-3

For x=2 . dy/dx=9

Since dy/dx is positve on the interval (-oo,-1) and (1,oo), and dy/dx is decreasing on interval (-1,1)

The function is increasing (-oo,-1) U (1,oo)

The function is decreasing (-1,1)