The problem reads: f(x)=7(e^2x)/x+4.

Ashlynn Hale 2022-08-14 Answered
What is the critical point of this function?
The problem reads: f ( x ) = 7 e 2 x x + 4..
I am unsure of how to approach this problem to find the derivative. If someone could break down the steps that would be greatly appreciated.
Also, the question asks for intervals of the increasing and decreasing parts of the function. How would I figure this out? I'm thinking I'd use a sign chart. But if you have any other useful methods, I am all ears, or rather eyes.
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Answers (1)

Cynthia Lester
Answered 2022-08-15 Author has 22 answers
Step 1
Let f ( x ) = 7 e 2 x x + 4 for x 0. Then, we have for x 0.
(1) f ( x ) = 7 e 2 x ( 2 x 1 x 2 )
where we used the product rule ( g h ) = g h + g h with g ( x ) = e 2 x , h ( x ) = 1 / x, g ( x ) = 2 e 2 x , and h ( x ) = 1 / x 2 .
Step 2
Now, we note from (1) that f ( x ) = 0 when x = 1 / 2.
Similarly, from (1), we see that when x > 1 / 2, f > 0 and f is increasing. When 0 < x < 1 / 2 or x < 0, f < 0 and f is decreasing.
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