Function value change with constrained arguments Suppose I have a function:
$$f(a,b)=h(a)+g(b)$$
where $a+b=1$. I'm interested in the values of $a$ and $b$ that maximize $f(a,b)$. 
Without substitution (e.g. by using the fact that $b=1-a$), how could I find out whether $f(a,b)$ is increasing in $a$?

My approach is as follows: since an increase in $a$ leads to a corresponding decrease in $b$, there are two effects of increasing $a$:
$$\frac{dh(a)}{da};-\frac{dg(b)}{db}$$
Such that the net effect is positive iff
$$\frac{dh(a)}{da}-\frac{dg(b)}{db}>0$$
Am I correct?
 A: You can find the restricted local extrema with Lagrange multipliers:
The local extrema of
$$f(x,y) = h(x) + g(y)$$
with the restriction
$$k(x,y) = x + y = 1$$
are given by
$$\nabla f(x,y) = \lambda\nabla k(x,y).$$
In the case:
$$(h'(x),g'(y)) = \lambda(1,1),$$
i.e.,
$$h'(x) = g'(y).$$
About your second question, the derivative of
$$x\longmapsto h(x) + g(1 - x)$$
is $$h'(x) - g'(1 - x).$$
Then if
$$h'(x) - g'(y) > 0,\qquad x + y = 1$$
it will be increasing, but the iff is false as the Henry's comment proves.
A: More generally, let $a$ and $b$ be functions of $t$ ($t$ could be $a$ or $b$ or something else entirely). Then
$$
\frac{\mathrm{d}}{\mathrm{d}t}f(a,b)=h'(a)\frac{\mathrm{d}a}{\mathrm{d}t}+g'(b)\frac{\mathrm{d}b}{\mathrm{d}t}
$$
Since $a+b=1$, we have $\frac{\mathrm{d}b}{\mathrm{d}t}=-\frac{\mathrm{d}a}{\mathrm{d}t}$. Therefore,
$$
\frac{\mathrm{d}}{\mathrm{d}t}f(a,b)=(h'(a)-g'(b))\frac{\mathrm{d}a}{\mathrm{d}t}
$$
This means that if $a$ is increasing, so that $\frac{\mathrm{d}a}{\mathrm{d}t}\gt0$, $f(a,b)$ will be increasing if
$$
h'(a)-g'(b)\gt0
$$
So your approach appears reasonable.
A: If $(a,b)$ is a feasible point then the feasible points nearby are $(a+t,b-t)$ with $|t|\ll1$, and we have
$$\eqalign{f(a+t,b-t)-f(a,b)&=\bigl(h(a+t)-h(a)\bigr)+\bigl(g(b-t)-g(b)\bigr)\cr &=t\>\bigl(h'(a)-g'(b)+o(1)\bigr)\> \qquad(t\to0) .\cr}$$
If $h'(a)-g'(b)\ne0$ then we can make the RHS positive with a suitable choice of $t$, hence $f$ cannot be maximal at the point $(a,b)$. It follows that $h'(a)=g'(b)$ is a necessary condition for a (conditional) local maximum of $f$ at $(a,b)$.
The above argument identifies the conditionally stationary points. Wether we have a minimum, maximum, or neither, in such a point has to be analyzed separately, e.g.,  by computing second derivatives, or through "global arguments", like convexity, values at the boundary points, etc.
