📂Analysis

The Relationship between Derivatives and the Increasing/Decreasing of Functions

정리

Let the function $f$ be differentiable at $(a,b)$.

• If for all $x\in (a,b)$, $f^{\prime}(x) \ge 0$ holds, then $f$ is monotonically increasing.

• If for all $x\in (a,b)$, $f^{\prime}(x)=0$ holds, then $f$ is a constant function.

• If for all $x\in (a,b)$, $f^{\prime}(x) \le 0$ holds, then $f$ is monotonically decreasing.

Proof

From the Mean Value Theorem, it follows that for all $x_{1},x_{2}\in (a,b)$ and $x \in (x_{1},x_{2})$ the following holds.

$$f(x_{2}) - f(x_{1})=(x_{2}-x_{1})f^{\prime}(x)$$

Since $x_{2}-x_{1}>0$, if $f^{\prime}(x)\ge 0$, then $f(x_{2})-f(x_{1})\ge 0$, which means that $f$ is a monotonically increasing function.

The same applies to the other cases.

■