Exercise 617Calculus › Single-variable calculus › Differentiation › Mean Value TheoremL1Added 4 Nov 2026

Sign of the derivative and monotonicity

#monotonicity#derivative sign#consequence

Statement

Show that if $f'(x)\ge0$ on $]a,b[$ , then $f$ is nondecreasing on $[a,b]$ .

Hint

Apply the MVT to $x<y$ in $[a,b]$ .

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MVT. For $x<y$ , there is $c\in]x,y[$ with $f(y)-f(x)=f'(c)(y-x)\ge0$ .

Conclusion. $f(y)\ge f(x)$ : $f$ is nondecreasing.

Strict version. If $f'>0$ , the same argument shows that $f$ is strictly increasing.