Calculus. Please show work

Question

Let 𝑓 be the function given by 𝑓(𝑥) = 3𝑒^2x and let 𝑔 be the function given by 𝑔(𝑥) = 6𝑥³. At what value of 𝑥 do the graphs of 𝑓 and 𝑔 have parallel tangent lines?

Answer 1

Given:

f(x) = 3 e^2x

g(x) = 6 x³

Find:

What x-value do f and g have parallel tangent lines?

f '(x) = g '(x), x = ?

Solution:

6 e^2x = 18 x²

e^2x = 3 x²

e^2x - 3 x² = 0

( e^x )² - ( x √( 3 ) )² = 0

( e^x - x √( 3 ) )( e^x + x √( 3 ) ) = 0

e^x - x √( 3 ) = 0

e^x = x √( 3 )

1 / √( 3 ) = x e^-x

-x e^-x = -1 / √( 3 )

Sort of similar to how the natural log, ln, undoes exponential, e, this is what the Lambert W function does:

x = y e^y ⇒ y = W( x )

The MASSIVE requirement is that the "coefficient" on e HAS to be EXACTLY equal to the POWER OF e

There are a LOT of other bounds and requirements that you can find on its Wiki page if you search it on Wikipedia.

-x = W( -1 / √( 3 ) )

x = - W( -1 / √( 3 ) ); however, there is a bound restriction that the argument in the Lambert W function where it cannot be less than - 1 / e, therefore, this is an extraneous solution

e^x + x √( 3 ) = 0

e^x = -x √( 3 )

1 / √( 3 ) = -x e^-x

-x = W( 1 / √( 3 ) )

x = - W( 1 / √( 3 ) ) ≈ -0.390646

I hope this helps! Please message me in the comments if you have any questions, comments, or concerns!

Answer 2