# Homework Help: Slope of normal to a given function's inverse

1. Apr 22, 2010

### Elixer

1. The problem statement, all variables and given/known data
f(x) = 2x2 + 4e(5x)
is invertible. Give the slope of the normal line to the graph of f-1 at x = 4.

2. Relevant equations
(Given in question)

3. The attempt at a solution
I don't know how to solve this question. But , I found the following:-

f(4) = 32 + 4e20

let y = f-1(x), then
dy/dx = 1/f ' (y)
f ' (x) = 4x + 20e(5x)
Hence, dy / dx = 1/(4y + 20e(5y))
I don't know if I am heading in the right direction.
Thank you.

2. Apr 22, 2010

### LCKurtz

Actually,the function is not invertible because it isn't 1-1. But you might observe that f(0) = 4 and it is invertible in a neighborhood of x = 0.

But f(4) isn't relevant to the question. The question is about the inverse function's slope at x = 4. Remember that f(0) = 4 means (0,4) is on the graph of y = f(x) and it also means that f-1(4) = 0 and (4,0) is on the graph of y = f-1(x).

The slope of that function at that point is what you are looking for. Does that help?

3. Apr 23, 2010

### Elixer

So, (4,0) lies on the graph of f-1(x),
y = f-1(x)
dy / dx = 1/(4y + 20e(5y))
Hence , slope of f-1(x) , dy / dx = 1/20
which implies , slope of the normal to the graph = -1 /(1/20) = -20
Ans = -20

Am I correct?
Thanks a lot!

4. Apr 23, 2010

### LCKurtz

If you calling y = f-1(x) then I would write

y' = 1 / f'(x) = 1/(4x + 20e(5x))