# Differentiating a parametrized piecewise function

Here's my piecewise function in which $\eta$ is a parameter: \begin{align} h(\eta)=\begin{cases} f(x) & x\leq g(\eta) \\ 0 & x>g(\eta) \end{cases} \end{align} where $x$ is a positive continuous strictly increasing function of $\eta$, and $f,g,h,$ are positive continuous strictly increasing functions of their arguments. I wish to find the extremum of function $h$, for which I need $dh/d\eta$. I don't know how to even proceed with this one. This equation actually comes from an optimization problem modeling a physical system. Any help is appreciated. Thanks in advance.

Since it is given that $h$ is positive, the inequality $x>g(\eta)$ can never hold. Thus $h(\eta)=f(x)$. Does this help?
• So what you are saying is $\frac{dh}{d\eta}=\frac{df}{dx}\frac{dx}{d\eta}$? – Deep Apr 25 '17 at 8:58
• Yes, this should be the case. But if you just want to find the extremum of $h$, note that $f$ is strictly increasing in $x$, i.e. to increase $f$ is to make $x$ large. But as $x$ is always strictly increasing in $\eta$, I think $h$ can only be maximized (resp. minimized) when $\eta \to +\infty$ (resp. $-\infty$). In other words, there's no such local extremum. – delt31 Apr 25 '17 at 9:05
• But it all depends on the fact that $h$ is positive. – delt31 Apr 25 '17 at 9:12