Critical points and extrema

A function plotted with its first and second derivatives. Find the critical points, classify them, and find the extrema on a closed interval.

Figure 1.3.Choose a function, then move the probe x₀ to read f′ and f″. Tick the interval box and drag p and q to see where the global maximum and minimum fall. Open on its own

A critical point is only a candidate. Whether it is a maximum, a minimum or neither depends on how the sign of f′ changes around it.

A squared factor in f′ does not change sign, so it gives a critical point that is not an extremum.