StellarThe Second Derivative Test is a shortcut for classifying a critical point when f'(c)=0 and the second derivative exists there. If f''(c)>0, then the graph is concave up at c, so f(c) is a local minimum. If f''(c)<0, then the graph is concave down at c, so f(c) is a local maximum. The intuition is that a concave up graph bends upward like a bowl, so the critical point sits at the bottom, while a concave down graph bends downward, so the critical point sits at the top. This test is useful because it avoids making a full sign chart for f'(x) in many cases.
However, the test does not always work. If f''(c)=0, then the test is inconclusive, and the point could be a maximum, minimum, or neither. In that case, you must use another method, usually the First Derivative Test. So the Second Derivative Test is best understood as a quick classification tool that works under the right conditions, not as a universal method.