4.3 | Increasing and Decreasing Functions and the First Derivative Test


Test for Increasing and Decreasing Functions
Let \(f\) be a function that is continuous on the closed interval \([a,b]\) and differentiable on the open interval \((a,b)\).

  1. If \(f'(x) >0\) for all \(x\) in \((a,b)\), then \(f\) is increasing on \([a,b]\).
  2. If \(f'(x) < 0\) for all \(x\) in \((a,b)\), then \(f\) is decreasing on \([a,b]\).
  3. If \(f'(x) = 0\) for all \(x\) in \((a,b)\), then \(f\) is constant on \([a,b]\).
The First Derivative Test
Let \(c\) be a critical number of a function \(f\) that is continuous on an open interval \(I\) containing \(c\). If \(f\) is differentiable on the interval, except possibly at \(c\), then \(f(c)\) can be classified as follows.

  1. If \(f'(x)\) changes from negative to positive at \(c\), then \(f\) has a relative minimum at \((c, f(c))\).
  2. If \(f'(x)\) changes from positive to negative at \(c\), then \(f\) has a relative maximum at \((c, f(c))\).
  3. If \(f'(x)\) doesn’t change signs at \(c\), then \(f(c)\) is not a relative extrema.

E 4.3 Exercises