Question

The function f(x)=(3x−3)e−6x has one critical number. Find it.

Answer 1

is the function this?
\[\Large f(x) = (3x-3)e^{-6x}\]

Answer 2

ya

Answer 3

Use the product rule to get
\[\Large f(x) = (3x-3)e^{-6x}\]
\[\Large \frac{d}{dx}[f(x)] = \frac{d}{dx}[(3x-3)e^{-6x}]\]
\[\Large f \ '(x) = \frac{d}{dx}[(3x-3)]e^{-6x}+(3x-3)\frac{d}{dx}[e^{-6x}]\]
\[\Large f \ '(x) = 3e^{-6x}+(3x-3)*(-6e^{-6x})\]
\[\Large f \ '(x) = 3e^{-6x}-18xe^{-6x}+18e^{-6x}\]
\[\Large f \ '(x) = -18xe^{-6x}+21e^{-6x}\]
\[\Large f \ '(x) = -3e^{-6x}(6x-7)\]

Answer 4

Now plug in f ' (x) = 0 and solve for x


\[\Large f \ '(x) = -3e^{-6x}(6x-7)\]
\[\Large 0 = -3e^{-6x}(6x-7)\]


I'll let you take over from here

Answer 5

still not getting it

Answer 6

Do you agree with how I got \(\Large f \ '(x)\) ?

Answer 7

yes

Answer 8

and you agree that we set f ' (x) equal to zero? or no?

Answer 9

\[\Large f \ '(x) = -3e^{-6x}(6x-7)\]
\[\Large 0 = -3e^{-6x}(6x-7)\]
\[\Large -3e^{-6x}(6x-7) = 0\]
\[\Large -3e^{-6x} = 0 \ \text{ or } \ 6x-7 = 0\]


The equation \[\Large -3e^{-6x} = 0\] has no solutions.


The equation \[\Large 6x-7 = 0\] has the solution x = _______ (fill in the blank)

Answer 10

thanks!

Answer 11

np