Question

The derivative of a function f is given by f'(x)= (x-3)e^x for x>0 and f(x)=7? a.) The function has a critical point at x=3. At this point, does f have a relative minimum or neither?

Answer 1

The function `f(x)` has a local maximum at `x=3`. See explanation.

Explanation:

To find if a function has a critical point at a place where `f'(x)=0` you have to check if the derivative changes sign at this point. If the change occurs then #f(x) has:

• Minimum if `f'(x)` changes sign from negative to positive
• Maximum if `f'(x)` changes sign from positive to negative.

To check it you can calculate the second derivative:

`f''(x)=1*e^x-(x-3)e^x=(1-x+3)e^x=(2-x)e^x`

`f''(3)=(2-3)e^3=-e^3<0`

`f''(x)` is negative in `x=3`. This means that `f'(x)` is decreasing at `x=3`, this finally means that `f(x)` has a MAXIMUM at `x=3`