calculus function question

Question

The factored forms of f'(x) and f''(x) are shown below.

f '(x) = e^(−x)[ (x − 3)^2 (x − 5)]

f ''(x) = = −e ^(−x) [ (x − 7)(x − 3)(x − 4)]

interval(s) where f(x) is increasing: ??

interval(s) where f(x) is decreasing: ??

local minimum(s): local maximum(s): ??

interval(s) where f(x) is concave up: ??

interval(s) where f(x) is concave down: ??

inflection point(s): ??

the e in both derivatives is confusing to me. For increasing/decreasing I found that f(x) using the first derivative, increases on the interval 5 < x , and decreases on x < 3, and increases on 3 < x < 5 (not sure about this one) and I don't know what to do with the e. the rest I am very unsure how to do.

Answer 1

my mistake, it checks out that f'' is derivative of f' for problem as given

look at the f' equality- it can only be negative if x<5 since (x-3) is squared

so f is decreasing when x<5, and increasing x>5

by observation f'=0 x=3 and x=5.

at x=3, f''=0 so that is an inflection point.

at x=5 f''>0 meaning there is a local minimum

(second derivative test)

for f'', if x<3, four negatives multiplied is positive

for 3<x<4, three negatives multiplied is negative

for 4<x<7, two negatives multiplied is positive

for x>7 f'' is negative

f is concave up x<3

f is concave down 3<x<4

f is concave up 4<x<7

f is concave down x>7