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Proposition: Characterization of Monotonic Functions via Derivatives

Let $a < b,$ $[a,b]$ be a closed real interval, and $f:[a,b]\to\mathbb R$ a continuous function. Further, let $f$ be differentiable on the open real interval $]a,b[.$

The function $f$ is:

• strictly monotonically increasing if $f'(x) > 0,$
• monotonically increasing if $f'(x) \ge 0,$
• monotonically decreasing if $f'(x) \le 0,$
• strictily monotonically decreasing if $f'(x) < 0,$

for all $x\in]a,b[.$

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983