If the eigenvalues are all negative, then x is a local maximum, and if some are positive and some negative, then the point is a saddle point. If the Hessian matrix is singular, then the second-derivative test is inconclusive.

How do you know if the second derivative test is inconclusive?

If f′(c)=0 and f″(c)=0, or if f″(c) doesn't exist, then the test is inconclusive.

What does Second derivative test tell you?

By taking the derivative of the derivative of a function f, we arrive at the second derivative, f′′. The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to f is increasing or decreasing.