
Hyperbolic Functions
Consider $\displaystyle f(x) = asinhx + bcoshx$ where a and b are real numbers.
For what values of a and b does f have an extremum at x = 0?
For what values of a and b does f have an extremum at x = c?
I started out by differentiating:
$\displaystyle f'(x) = acoshx + bsinhx$
At extremum f'(x) = 0. So setting f'(x) = 0 and x = 0,
$\displaystyle 0 = a*1 + b*0 $
? Doesn't seem to be correct.
I don't know where to go from here to solve this problem. If anyone could help me out, would be greatly appreciated, I don't understand the overall concept of hyperbolic functions very well. Thanks!

Set f'(x) = 0 and solve for a, b given x.
x=0: $\displaystyle a(1)+b(0)=0 \implies a=0, b \in R$
At a general point x=c:
$\displaystyle a\cosh{c} + b\sinh{c}=0$
$\displaystyle \Rightarrow a\cosh c = b\sinh c$
$\displaystyle \Rightarrow a = b\frac{\sinh c}{\cosh c} = b\tanh c, ~b\in R$

Thanks a bunch! At least I was on the right track with this question, that's a relief.
Would you mind explaining the last point a bit more though?
What I'm confused about is:
$\displaystyle a = btanhc $ and how that goes to b is any real number.

$\displaystyle \sinh{x}/\cosh{x}=\tanh x$.
And $\displaystyle b\in R$ means b is any real number. (I just solved for a in terms of b, since there are infinitely many solutions)
The solution to the equation $\displaystyle f'(c) = 0$ can be expressed as a set $\displaystyle S=\{ (a,b) \mid a=b\tanh{c}, ~b\in\mathbb{R}\}$.
I hope that's right.

I am unsure if the term "extremum" refers to local or abolute min, I don't know if it applies, however is it sufficient to only set the derivative to equal zero for extremum or will the second derivative test need to be used to verify that these are not points of inflection?

You are correct! You must ensure that the second derivative is not 0, otherwise you don't (necessarily) have an extremum.
An extremum is any local or global max or min.