What you’ll learn to do: Evaluate the graph of a function using the first and second derivative test

Earlier in this module we stated that if a function [latex]f[/latex] has a local extremum at a point [latex]c[/latex], then [latex]c[/latex] must be a critical point of [latex]f[/latex]. However, a function is not guaranteed to have a local extremum at a critical point. For example, [latex]f(x)=x^3[/latex] has a critical point at [latex]x=0[/latex] since [latex]f^{\prime}(x)=3x^2[/latex] is zero at [latex]x=0[/latex], but [latex]f[/latex] does not have a local extremum at [latex]x=0[/latex]. Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.