Subtracting convex function from concave function

Question

Is a function $h(x)=af(x)-g(x)$ convex if $a$ is positive, $f(x)$ concave and $g(x)$ convex? How and where to start the proof?

Answer 1

$h$ is not convex. It is concave: let $0<t<1$. Then

$f(tx+(1-t)y) \geq tf(x)+(1-t)f(y)$ and

$g(tx+(1-t)y) \leq tg(x)+(1-t)g(y)$.

Multiply the first equation by $a$ the second one by $-1$ and add. Remember that when you multiply an inequality by $-1$ $\leq $ becomes $\geq$. Now you get

$(af-g)(tx+(1-t)y) \geq t(af-g) (x)+(1-t)(af-g)(y)$.