Consider the function x^(2)e^(-x) for -1≤x≤3.

Question

a. For what values of x is f increasing? Decreasing?

b. Find the values of x where f(x) is the greatest; where f(x) is the least.

c. Find the values of x where f is increasing most rapidly; where f is decreasing most rapidly.

d. Sketch the graph of f, labeling your answers from parts (a-c).

 

Answer 1

f(x)=x²e^-x, -1≤x≤3

a)f'(x)=(2x-x²)e^-x=0 for x=0 and x=2.

f''(x)=(2-4x+x²)e^-x

f''(0)=2>0-->x=0 is min

f''(2)=-2/e²<0 -->x=2 is max.

a) So, f(x)=x²e^-x is increasing in the interval (0, 2) and is  decresing in (-1,0) and (2,3).

b) x=0 is min, f(0)=0,which is minimum value of f(x)

 for x=2, f(2)=4/e², maximum value of f in (-1, 3)

c) look at the sign of f''(x).

d) sketch graph yourself