# Prove that e^c = sinh(1) for some c in [-1,1]

ruby_duby

## Homework Statement

Assuming that (ex)’ = (ex) at all point x, prove that ec
= sinh(1) for some point C $$\in$$ (-1,1). Give the full proof.

## The Attempt at a Solution

I honestly don't know how to attempt this. i can show that (ex)’ = (ex) by:

(ex) = 1 + x + x2$$/2!$$ + x3$$/3!$$ + x4! + x5$$/5!$$ +...

(ex)’ = 1 + x + x2$$/2!$$ + x3$$/3!$$ + x4! + x5$$/5!$$ +...

I just don't know how to tackle the sinh part of the equation.

I would really appreciate any help/ guidance

Billy Bob

Don't try series. Try Mean Value Theorem.

And don't forget the definition of sinh.

ruby_duby

thanks so much, I've managed to work out the answer, however, does anyone know if i need to find a specific value for c?

Billy Bob

Usually you don't, but if you do it's easy enough:

e^c=sinh(1), so c=ln(sinh(1))