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 sec...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.
let \(f\) be a continuous function over an interval \(I\) containing a critical point \(c\) such that \(f\) is differentiable over \(I\) except possibly at \(c\); if \(f'\) changes sign from positive ...let \(f\) be a continuous function over an interval \(I\) containing a critical point \(c\) such that \(f\) is differentiable over \(I\) except possibly at \(c\); if \(f'\) changes sign from positive to negative as \(x\) increases through \(c\), then \(f\) has a local maximum at \(c\); if \(f'\) changes sign from negative to positive as \(x\) increases through \(c\), then \(f\) has a local minimum at \(c\); if \(f'\) does not change sign as \(x\) increases through \(c\), then \(f\) does not hav…
