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

Question

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?

S. Iason Koutsoulis · Accepted Answer

If you remember, in a two-dimensional curve, its concavity or convexity (the slope of its slope) is given by the second derivative. For a three-dimensional function, you want to look at the Hessian table (the table of all second derivatives).
If the Hessian is negative definite for all values then the function is strictly concave, and if the Hessian is positive definite for all values then the function is strictly convex. If the Hessian is not negative semidefinite for all values then the function is not concave, and hence of course is not strictly concave. So on, so forth.
This particular function will be a bit more tricky to find second derivatives, I presume.
There are various examples shown in a very nice website here.
There is even a Cobb-Douglas example, which I'm certain you'll find valuable.

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

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