Exercise 1271Calculus › Single-variable calculus › Differentiation › Mean Value TheoremL1Added 4 Nov 2026

Bounding $f(b)-f(a)$

#MVT#bound#interval

Statement

Let $f$ be $C^1$ on $[0,1]$ with $|f'|\le2$ . Show that $|f(1)-f(0)|\le2$ .

Hint

Apply MVT.

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MVT. $f(1)-f(0)=f'(c)$ for some $c\in]0,1[$ , so $|f(1)-f(0)|\le2$ .

Conclusion. The inequality holds.