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

[ruby_duby]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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

(e^x) = 1 + x + x²$$/2!$$ + x³$$/3!$$ + x^4! + x⁵$$/5!$$ +...

(e^x)’ = 1 + x + x²$$/2!$$ + x³$$/3!$$ + x^4! + 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))

 

[CompuChip]

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?

You were only asked to show that it exists, not to actually give it (often in mathematics, it's easier to prove that you can construct or find it, than to actually do it).