f(x) = x^4 + ax^2

Question

What is a if f(x) has a local minimum at x=5.
How do I justify the answer.

The first derivative has to be zero at x = 5. You then check what the second derivative is at x = 5. If it is positive then that's ok.

In this case you find that a = -50 and that the second derivative at x = 5 is positive.