# How can I prove that a utility function does (or does not) satisfy diminishing MRS?

I have this CES utility function:

$$U(f, c) = (f^\alpha + c^\alpha)^{1/\alpha},$$

with $$\alpha > 0$$.

The problem set asks does it "satisfy the principle of diminishing marginal rate of substitution for any value of $$\alpha$$".

To which the answer is "False, if $$\alpha > 1$$ the indifference curves are concave, so they don’t satisfy the principle of diminishing MRS"

But short of graphing the indifference curves and hoping that it's visually obvious, I'm not sure how to tell this.

How can one tell from a utility function whether it is or is not non-concave?