Question

prove f(x) has an inverse function and find the slope of the tangent line at point P on the graph of f^-1 f(x) = 4x^5 - (1/(x^3)), x>0; P(3,1)

Answer 1

is this
\[f(x)=4x^5-\frac{1}{x^3}\]?

Answer 2

yes thats the correct equation :)

Answer 3

ok
do you know why it has an inverse?

Answer 4

one to one, constantly increasing

Answer 5

constantly increasing is not quite good enough, you have to also say it is continuous

Answer 6

in fact it is not continuous, but it is on \(x>0\) so you need to include that fact as well

Answer 7

sounds good, got that part down

Answer 8

now finding the slope is easier than it looks.
what you need to use is that
\[(f^{-1})'=\frac{1}{f'(f^{-1}(x))}\]\]

Answer 9

so you do not actually have to find the derivative of the inverse

Answer 10

\((1,3)\) is on the graph of \(f\) and so \((3,1)\) is on the graph of \(f^{-1}\)

Answer 11

so,
\[f' = 20x^4 -(3/x^4) + c\]
I am having a hard time finding the inverse thought

Answer 12

a couple mistakes there

Answer 13

but before we correct them, lets make sure one thing is clear
you do not need to find the inverse function
in fact, you probably cannot find it

Answer 14

hmm okay, that would probably makes a lot of sense considering my struggles over finding it

Answer 15

first off

\[f'(x)=20x^4+\frac{3}{x^4}\] your sign was wrong
also this is a derivative, not an integral, so there is no \(+C\) out at the end

Answer 16

clear enough, right?

Answer 17

oops, i read my paper wrong i had 20x^4 - -(3/x^4)

Answer 18

ok good
now look at the way to find the derivative of an inverse i wrote above, i will write it again
\[(f^{-1})'(x)=\frac{1}{f'(f^{-1}(x))}\]

Answer 19

but yes I understand, been a long day not sure whey I added the +C either :(

Answer 20

and so of course
\[(f^{-1})'(3)=\frac{1}{f'(f^{-1}(3))}\]

Answer 21

we know everything we need to know to compute this, without finding an explicit expression for \(f^{-1}(x)\) in fact, you can do it in your head

Answer 22

on the right hand side we need \(f^{-1}(3)\)
usually you have to find that, but not in this case because they told you the point is \((3,1)\) so you know \(f^{-1}(3)=1\)

Answer 23

so find \[1/ f'(1)\] ?

Answer 24

yup

Answer 25

like i said, you can do it in your head
\[f'(1)=20+3=23\] and so your answer is \(\frac{1}{23}\)

Answer 26

Thank you for your help and you time, I really appreciate it!

Answer 27

usually when people are given problems like this they spend 2 hours trying to find the inverse and get stuck, but notice how easy it is
yw

Answer 28

alot easier than i was expecting