Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing.
-$$f ^ { \prime } ( x ) = ( x + 2 ) ^ { 2 } e ^ { - x }$$
A) Decreasing on $( - \infty , - 2 )$; increasing on $( - 2 , \infty )$
B) Never decreasing; increasing on $( - \infty , - 2 ) \cup ( - 2 , \infty )$
C) Never decreasing; increasing on $( - \infty , 2 ) \cup ( 2 , \infty )$
D) Never increasing; decreasing on $( - \infty , - 2 ) \cup ( - 2 , \infty )$

Question

Quizplus · Accepted Answer

The derivative $f'(x) = (x + 2)^2 e^{-x}$ is always positive because $(x + 2)^2$ is always non-negative (and positive except at $x = -2$, where it is 0) and $e^{-x}$ is always positive for all real numbers $x$. Thus, $f(x)$ is always increasing, never decreasing.

Using the Derivative of F(x) Given Below, Determine the Intervals f′(x)=(x+2)2e−xf ^ { \prime } ( x ) = ( x + 2 ) ^ { 2 } e ^ { - x }f′(x)=(x+2)2e−x

Using the Derivative of F(x) Given Below, Determine the Intervals $f ^ { \prime } ( x ) = ( x + 2 ) ^ { 2 } e ^ { - x }$