Knowlpedia

Derivative sign implies monotonicity

A nonnegative derivative forces a function to be nondecreasing, and a nonpositive derivative forces it to be nonincreasing.

Derivative sign implies monotonicity: Let I⊆RI\subseteq\mathbb{R} be an , and let f:I→Rf:I\to\mathbb{R} be on II.

  • If f′(x)≥0f'(x)\ge 0 for all x∈Ix\in I, then ff is nondecreasing (monotone increasing) on II.
  • If f′(x)≤0f'(x)\le 0 for all x∈Ix\in I, then ff is nonincreasing (monotone decreasing) on II.
Remarks

This is proved by applying the on subintervals and interpreting the sign of the as controlling slopes. The strict version is , and the conclusion is a special case of being a .