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.