Second Derivative Test - Expii

The first derivative test gives one way to distinguish between saddle points and turning points, using the calculus of increasing and decreasing functions. Alternatively, you can use the second derivative test: a stationary point \(x=c\) of a function \(f\) must be a turning point if \(f\) is concave up or down at \(x=c\): if \(f''(c)\) is positive, then \(f(c)\) is a local minimum of \(f\), while if \(f''(c)\) is negative, then \(f(c)\) is a local maximum of \(f\). (However, the test is inconclusive if \(f''(c) = 0\).).