SOLUTION: Locate the critical points of the following function. Then use the second derivative Test to determine whether they correspond to local maxima, local minima, or neither. f(x)=-e^x

Algebra ->  Test -> SOLUTION: Locate the critical points of the following function. Then use the second derivative Test to determine whether they correspond to local maxima, local minima, or neither. f(x)=-e^x      Log On


   

Question 1155435: Locate the critical points of the following function. Then use the second derivative Test to determine whether they correspond to local maxima, local minima, or neither.
f(x)=-e^x (x-9)
Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


f%28x%29+=+-e%5Ex%28x-9%29+=+%289-x%29e%5Ex

Use the product rule to find the derivative:
df%2Fdx+=+%289-x%29e%5Ex-1%28e%5Ex%29+=+%288-x%29e%5Ex

The derivative is zero only at x=8; the function value at x=8 is %289-8%29e%5E8+=+e%5E8

The critical point is (8,e^8).

Use the product rule to find the second derivative:
d%5E2f%2Fdx%5E2+=+%288-x%29e%5Ex-1%28e%5Ex%29+=+%287-x%29e%5Ex

The value of the second derivative at x=8 is %287-8%29e%5E8+=+-e%5E8

The second derivative is negative at x=8; the critical point is a maximum.

graph%28400%2C400%2C-2%2C12%2C-1000%2C5000%2C%289-x%29e%5Ex%29