1. ## Concave function

Let f(x) be a concave function of N variables defined over a convex set,S $\in R^N$Let g be an increasing and concave function of one var.

prove that g(f(x)) is concave over S

2. Probably obvious, but,...

It seems the second derivative should get us somehwere...

$g''(f(x)) \cdot f'(x) + g'(f(x)) \cdot f''(x)$

The information in the problem statement, relating to g'' and f'' is too compelling to pass up. Also, with f' positive, it seems we've only g' to argue. Will the convexity of S do that?

For what it's worth...

3. Originally Posted by TKHunny
Probably obvious, but,...

It seems the second derivative should get us somehwere...

$g''(f(x)) \cdot f'(x) + g'(f(x)) \cdot f''(x)$

The information in the problem statement, relating to g'' and f'' is too compelling to pass up. Also, with f' positive, it seems we've only g' to argue. Will the convexity of S do that?

For what it's worth...
Thanks! But the fact that f(x) is a function of many variables would't force us to evaluate the second derivative with the gradient of f? In other words remember that x is a vector

4. Originally Posted by altave86
Let f(x) be a concave function of N variables defined over a convex set,S $\in R^N$Let g be an increasing and concave function of one var.

prove that g(f(x)) is concave over S
The proof by TKHunny is actually not true since we don't know that the functions are differentiable.
However, as he mentioned the proof is obvious.
Let me give it below.
Proof. Since $f:\mathbb{R}^{n}\to\mathbb{R}$ is concave, we have
$f\big(\lambda x+(1-\lambda)y\big)\geq\lambda f(x)+(1-\lambda)f(y)$ for all $x,y\in\mathbb{R}^{n}$ and all $\lambda\in[0,1]$. (*)
As $g:\mathbb{R}\to\mathbb{R}$ is increasing, we use (*) to get
$g\Big(f\big(\lambda x+(1-\lambda)y\big)\Big)\geq g\big(\lambda f(x)+(1-\lambda)f(y)\big)$.........( $g$ is increasing)
............................... $\geq \lambda g\big(f(x)\big)+(1-\lambda)g\big(f(y)\big)$.....( $g$ is concave)
for all $x,y\in\mathbb{R}^{n}$ and all $\lambda\in[0,1]$.
This completes the proof that $g\circ f$ is concave on $\mathbb{R}^{n}$..... $\rule{0.3cm}{0.3cm}$

5. I dare you to show me where I offered a proof. I just said it was compelling, not useful.

I often can be found not knowing that I have no idea what I'm talking about.