find the derivative at 12 of the inverse of the function f[x] = .25(x^5 + 2x^3)

Question

anonymous · Answer

I know so far that i use 1/f'[f^-1[x]]. However, I do not know how to apply this

anonymous · Answer

Did you say something tkhunny?

tkhunny · Answer

You have f(x).  Find f'(x).

anonymous · Answer

ok f'[x] = .25(6x^2 + 5x^4)


Now what?

tkhunny · Answer

Well, there is little hope if finding $f^{-1}(x)$ explicitly, so this must be a numerical problem.  This is fine because we have x = 12 as the place where we are interested.

anonymous · Answer

I tried using the definition of an inverse (domain and range flipped), but there is NO way to solve that equation. I am desperate

tkhunny · Answer

You do not need to solve for the inverse.  You need to evaluate the inverse at x = 12.

f(2) = 12  --  Serendipity provides that this is the ONLY Real Solution, so there is no ambiguity.

$f^{-1}(12) = 2$

tkhunny · Answer

So, $f'(f^{-1}(12)) = f'(2)$.  Can you take it from there?

anonymous · Answer

Yes I can easily take it from there. However, can you elaborate on how you got 2? I don't see any hint in the problem that would let us find 2 in there

tkhunny · Answer

Well, somehow, you just have to solve for it.

$\dfrac{1}{4}\left(x^{5}+2x^{3}\right) = 12$

$\left(x^{5}+2x^{3}\right) = 48$

$x^{5}+2x^{3} - 48 = 0$

The HAS a positive solution in the Real Numbers (do you know why?).  If it's Rational, which I hope it is, it will be a factor of 48 (do you know why?).  This narrows the choices considerably.

anonymous · Answer

i'm not sure why we know why it has a real positive solution, but I do know why it has a factor of 48. Can you elaborate on that?

tkhunny · Answer

Perhaps you have heard of "Descartes' Rule of Signs"?