Answer to Question #17682 in Calculus for hsd

Question

Consider the function f(x)=x^2e^(5x). For this function there are three important intervals: (−∞,A], [A,B], and [B,∞) where A and B are the critical numbers. Find A and B.

A=
B=

For each of the following intervals, tell whether f(x) is increasing (enter INC ) or decreasing (enter DEC ).

(−∞,A]:
[A,B]:
[B,∞)

Answer 1

first we find derivative. Hereit is
2xe^(5x)+5x^2e^(5x)
Easy to see that there two zeros
x=0 and x=-2/5
so A=-2/5 and B=0.
to see inc and dec regions we have just to check value of derivative. If it is
positive at certain region then we have inc, if negative - dec. Hence
(−∞,A]:inc[A,B]:dec
[B,∞):inc