Consider the function f(x)=x^(2)e^(17x). For this function there are three important intervals: (-inf,A],[A,B], and [B,inf) where A and B are the critical numbers with A<B.

Question

(inf represents infinity)

A = ?

and

B = ?

Answer 1

A = - 2/17

B = 0

f(x) increases oi interval (-inf,-2/17) and f(x) decreases in interval (-2/17, 0) then f(x) increase in interval (0, +inf)

Answer 2

To find critical numbers, find the derivative and set it = 0, then solve for the x's.

For this function one must use the product rule. So the derivative is

f '(x) = x²[e^17x (17)] + e^17x(2x)

Factor out an e^17x and set the derivative = 0

0 = e^17x [ 17x² + 2x]

e^17x will never = 0, so we concentrate on

17x² + 2x = 0

x(17x + 2) = 0

Critical numbers found where x = 0 and

where 17x+2 = 0

So the critical numbers are -2/17 (A) and 0 (B)