Question

Consider the following function. f(x) = (3 − x)e−x (a) Find the intervals of increase or decrease. a. increasing b. decreasing(b) Find the intervals of concavity. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) a. concave up b. concave down(c) Find the point of inflection. (If an answer does not exist, enter DNE.)(x, y) =

Answer 1

Answer:

a)

f is increasing in the interval $(4,+\infty)$

f is decreasing in the interval  $(-\infty,4)$

b)

f is concave up in the interval $(-\infty,5)$

f is concave down in the interval  $(5,+\infty)$

c)

x = 5 is the point of inflection.

Step-by-step explanation:

We have

$f(x)=(3-x)e^{-x}$

(a) Find the intervals of increase or decrease.

To find where the function is increasing, we must find the interval(s) where f'(x)>0

By the rule of the derivative of a product:

$f'(x)=-e^{-x}-(3-x)e^{-x}\\\\f'(x)>0\Rightarrow -e^{-x}>(3-x)e^{-x}$

since

$e^{-x}>0$

we divide both sides by

$e^{-x}>0$

and we get

$-1>3-x\Rightarrow x>4$

so f is increasing in the interval  

$(4,+\infty)$

Similarly, we can see f is decreasing in the interval

$(-\infty,4)$

(b) Find the intervals of concavity.

The function is concave up in the interval(s) where f''>0

$f''(x)=2e^{-x}+(3-x)e^{-x}\\\\f''(x)>0\Rightarrow 2e^{-x}+(3-x)e^{-x}>0\Rightarrow 2+3-x>0\Rightarrow x<5$

so f is concave up in the interval

$(-\infty,5)$

Similarly, we can see f is concave down in the interval

$(5,+\infty)$

(c) Find the point of inflection.

Since f changes its concavity at x=5, this point is a point of inflection.