4.1 | Using First and Second Derivatives


Suppose \(f\) is defined on an interval and has a local maximum or minimum at \(x = a,\) which is not an endpoint of the interval. If \(f\) is differentiable at \(x=a,\) then \(f'(a) = 0.\) Thus, \(a\) is a critical point.
The First-Derivative Test for Local Maxima and Minima:
Suppose \((p, f(p))\) is a critical point of a continuous function \(f.\) Moving from left to right:

  • If \(f’\) changes from negative to positive at \(p,\) then \(f\) has a local minimum at \(p.\)
  • If \(f’\) changes from positive to negative at \(p,\) then \(f\) has a local maximum at \(p.\)
The Second-Derivative Test for Local Maxima and Minima:

  • If \(f'(p) = 0\) and \(f^{\prime \prime }(p) > 0,\) then \(f\) has a local minimum at \(p.\)
  • If \(f'(p) = 0\) and \(f ^{\prime \prime } (p) < 0,\) then \(f\) has a local maximum at \(p.\)
  • If \(f'(p) = 0\) and \(f^{\prime \prime } (p) = 0,\) then the test is inconclusive.
Suppose \(f^{\prime \prime }\) is defined on both sides of \(p\):

  • If \(f^{\prime \prime } \) is zero or undefined at \(p,\) then \((p,f(p))\) is an inflection point.
  • To test whether \((p,f(p))\) is an inflection point, check whether \(f^{\prime \prime } \) changes sign at \(p.\)

E 4.1 Exercises