1. increasing and decreasing functions

ok here is the problem, consider a function f whose derivative is given by f'(x) = (x-4)^2 * e^(-x/2).

a.) the function f is increasing on the interval?

b.) the function f is concave up on the interval(s)

c.) the function f is concave down on the interval(s)

ok so what i dont get is how to find the critical points. i only see one in the problem and thats 4.? right? and to find the concave up or down i would apply the second derivative test? this problem gets really messy once i try to do the 2nd derivative test. there has to be an easier way. can anyone help me out?

2. Originally Posted by slapmaxwell1
ok here is the problem, consider a function f whose derivative is given by f'(x) = (x-4)^2 * e^(-x/2).

a.) the function f is increasing on the interval?

b.) the function f is concave up on the interval(s)

c.) the function f is concave down on the interval(s)

ok so what i dont get is how to find the critical points. i only see one in the problem and thats 4.? right?

yes, for f'(x)

and to find the concave up or down i would apply the second derivative test? this problem gets really messy once i try to do the 2nd derivative test. there has to be an easier way. can anyone help me out?

yes ... $(x-4)^2 \cdot e^{-x/2} \ge 0$ for all x