I've been stuck on this problem for a while now because I'm unable to invert the function. Can someone please point me in the right direction 🙂
#Inverse Derivative
short peak
split dragon
What have you tried?
Because I don't actually think you should calculate it explicitly unless they question states you should? Otherwise use inverse function theorem
ornate blade
Because I don't actually think you should calculate it explicitly unless they qu...
What inverse function theorem?
split dragon
What inverse function theorem?
Perhaps I used the wrong name or am even wrong about what I remember. Isn't there a theorem stating something like the derivative of the inverse at a point is equal to the reciprocal of the composition of the derivative and inverse function at that point? Also need to be continuous but we have a polynomial which is obviously continuous
ornate blade
Perhaps I used the wrong name or am even wrong about what I remember. Isn't ther...
...okay, but you still need to know the inverse for that to work.
orchid cairn
I've been stuck on this problem for a while now because I'm unable to invert the...
Let φ(x) be the inverse to f(x). Then:
φ'(x) = 1/f'(φ(x))
So:
φ'(-14) = 1/f'(φ(-14))
So, we need to find φ(-14). For that, we solve f(x) = -14.
(1/27)(x^5 + 5x^3) = -14
x^5 + 5x^3 + 378 = 0
Using the integer root theorem, we eventually get that x = 3 is the solution.
So, φ(-14) = 3. The rest is easy.
ornate blade
Let φ(x) be the inverse to f(x). Then:
φ'(x) = 1/f'(φ(x))
So:
φ'(-14) = 1/f'(φ(-...
The solution? Wouldn't there be five solutions, since it's a quintic?
orchid cairn
*The* solution? Wouldn't there be five solutions, since it's a quintic?
Well, five complex - yes. But only one real.
ornate blade
Fair enough.