Suppose I wanted to show that a concave function $f:(a,b) \to \mathbb{R}$ which is $C^2$ must have negative second derivative at each $x\in (a,b)$. I might try this by finite difference, noting that if $f''(x)>0$, then $$\frac{f(x + \Delta x)+f(x-\Delta x)-2f(x)}{2(\Delta x)^2}$$
should be positive for small $\Delta x$, contradicting concavity.
Can you recommend another method for showing this besides finite difference? Is there a problem with my (admittedly sketchy) proof?
$$\frac{f(x + \Delta x)+f(x-\Delta x)-2f(x)}{2(\Delta x)^2} \le 0$$
The definition of concavity is that the value at the midpoint of any interval is less-than-or-equal-to the average of the values at the endpoints of the interval. $x$ is your midpoint, $x +\Delta x$ and $x - \Delta x$ are your endpoints.
By definition, $\frac {f(x + \Delta x)+f(x-\Delta x)} 2 \le f(x)$ which implies $f(x + \Delta x)+f(x-\Delta x) \le 2 f(x)$.
$$\frac{\text{some non-positive number}}{\text{some positive number}} \le 0$$