Question 7.T.9: Let f : [a, b] → R be a continuous function which is differe...

Elements of Real Analysis

Let $f : [a, b] → \mathbb{R}$ be a continuous function which is differentiable on (a, b).

(i) If $f^{′}(x) ≥ 0$ for all x ∈ (a, b), f is increasing on [a, b].

(ii) If $f^{′}(x) ≤ 0$ for all x ∈ (a, b), f is decreasing on [a, b].

(iii) If $f^{′}(x) > 0$ for all x ∈ (a, b), f is strictly increasing on [a, b].

(iv) If $f^{′}(x) < 0$ for all x ∈ (a, b), f is strictly decreasing on [a, b].

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