00:01
In part a, we determine the critical points of the function f.
00:05
We are given this derivative function f prime of x equals x plus 4 times e power negative x.
00:13
We find a critical point of f.
00:20
And for this, we set up the equation f prime of x equals 0 and we solve for x.
00:28
So let's replace f prime of x equals 0.
00:31
This gives 0 equals x plus 4 times e to the power negative.
00:37
Negative x solving this we get x plus 4 equals 0 or e to the power negative x equals 0 we get x equals negative 4 here e .0 negative x this is not 0 for any values of x so no solution or the solution does not exist so the only critical point of the function f occurs at x equals negative 4 this is the critical point of f.
01:19
In part b, we determine the increasing and decreasing intervals of f.
01:24
For that, we prepare the sign chart of f prime.
01:30
Sign chart of the derivative function f prime of x.
01:34
So let's put the number line and mark the only critical point, which is at x equals negative 4.
01:47
This is the sign chart for f prime of x.
01:50
Let's copy the derivative function x prime of x this equals x plus 4 times e to the power negative x.
02:01
Now we have two intervals over here that is negative infinity negative 4.
02:07
This is one interval and another interval is from negative 4 to infinity...