4.4 | Concavity and the Second Derivative Test

Definition of Concavity
Let \(f\) be differentiable on an open interval \(I\). The graph of \(f\) is concave upward on \(I\) when \(f’\) is increasing on the interval and concave downard on \(I\) when \(f’\) is decreasing on \(I\)
If \((c,f(c))\) is a point of inflection of the graph of \(f\), then either \(f^{\prime \prime}(c) = 0\) or \(f^{\prime \prime}\) does not exist at \(c\).
Test for Concavity
Let \(f\) be a function whose second derivative exists on an open interval \(I\).

  1. If \(f^{\prime \prime} (x) >0\) for all \(x\) in \((a,b)\), then \(f\) is concave upward on \(I\).
  2. If \(f^{\prime \prime} (x) < 0\) for all \(x\) in \((a,b)\), then \(f\) is concave downward on \(I\).
Definition of Point of Inflection
Let \(f\) be a function that is continuous on an open interval, and let \(c\) be a point in the interval. If the graph of \(f\) has a tangent line at this point \((c,f(c))\) and the graph of \(f\) changes concavity at \(c\), then the point \((c,f(c))\) is a point of inflection.
The Second Derivative Test
Let \(f\) be a function such that \(f'(c) = 0\) and the second derivative of \(f\) exists on an open interval containing \(c\).

  1. If \(f^{\prime \prime} (c) >0\), then \(f\) has a relative minimum at \((c,f(c))\).
  2. If \(f^{\prime \prime} (c) < 0\), then \(f\) has a relative maximum at \((c,f(c))\).

If \(f^{\prime \prime}(c) = 0\), then the test is inconclusive. That is, \(f\) may or may not have a relative extrema. In such cases, you can use the First Derivative Test.

E 4.4 Exercises