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Over what interval is the graph of f(x)=-(x-8)^2-1 decreasing

User DougW
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The graph of f(x)=−(x−8) ^2 −1 is decreasing on the interval x>8.

To find the interval over which the graph of the function

f(x)=−(x−8)^2 −1 is decreasing, you need to follow these steps:

Find the derivative of the function

f(x) with respect to x.

Set the derivative equal to zero and solve for

x to find critical points.

Use the first derivative test to determine where the function is increasing or decreasing.

Let's go through each step:

The Derivative

f(x)=−(x−8) ^2 −1

To find the derivative, apply the chain rule and the constant multiple rule:

f ′(x)=−2(x−8)(1)=−2(x−8)

Set the Derivative Equal to Zero and Solve for

−2(x−8)=0

Solve for

x:

x−8=0

x=8

So, there is one critical point at

x=8.

Use the First Derivative Test

Now, choose test points from each interval determined by the critical point

x=8 (e.g., x<8 and x>8).

Test point x=7 (from the interval x<8):

Substitute x=7 into f ′(7)=−2(7−8)=2 (positive)

This means that

f(x) is increasing on the interval

x<8.

Test point

x=9 (from the interval x>8):

Substitute x=9 into

f ′(9)=−2(9−8)=−2 (negative)

This means that f(x) is decreasing on the interval x>8.

Over what interval is the graph of f(x)=-(x-8)^2-1 decreasing-example-1
User Curlas
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