Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing.
-$$f ^ { \prime } ( x ) = ( x - 5 ) e ^ { - x }$$
A) Decreasing on $( - \infty , - 5 )$; increasing on $( - 5 , \infty )$
B) Increasing on $( - \infty , 5 )$; decreasing on $( 5 , \infty )$
C) Increasing on $( - \infty , - 5 )$; decreasing on $( - 5 , \infty )$
D) Decreasing on $( - \infty , 5 )$; increasing on $( 5 , \infty )$

Question

Quizplus · Accepted Answer

To determine the intervals on which a function is increasing or decreasing, we need to look at the sign of the derivative. If the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing. From the given derivative, we can see that it factorizes as:

$$f ^ { \prime } ( x ) = ( x - 5 ) e ^ { - x } = 0$$

This equation has a root at $x=5$, which divides the real line into two intervals: $( - \infty , 5 )$ and $( 5 , \infty )$. We can then evaluate the sign of the derivative in each of these intervals by picking a test point, plugging it into the derivative and checking its sign. For example, for $x=0$, we have:

$$f ^ { \prime } ( 0 ) = ( 0 - 5 ) e ^ { - 0 } = -5 < 0$$

This means that the function is decreasing at $x=0$, which is in the interval $( - \infty , 5 )$. Repeating this method for any test point in this interval will give the same result, and we can conclude that the function is decreasing in $( - \infty , 5 )$.

Similarly, for any test point in $( 5 , \infty )$, we get a positive value for the derivative, and therefore, the function is increasing in $( 5 , \infty )$.

Therefore, the best choice is D: Decreasing on $( - \infty , 5 )$; increasing on $( 5 , \infty )$.

Using the Derivative of F(x) Given Below, Determine the Intervals f′(x)=(x−5)e−xf ^ { \prime } ( x ) = ( x - 5 ) e ^ { - x }f′(x)=(x−5)e−x

Using the Derivative of F(x) Given Below, Determine the Intervals $f ^ { \prime } ( x ) = ( x - 5 ) e ^ { - x }$