Question Number 122293 by mathocean1 last updated on 15/Nov/20
![given f(x)=cos^2 x for x ∈ [−(π/(12));(π/(12 ))] , ∣f′(x)∣≤(1/2). 1) show that ∀ x, y ∈ [−(π/(12));(π/(12))] ; ∣cos^2 x−cos^2 y∣≤(1/2)∣x−y∣](https://www.tinkutara.com/question/Q122293.png)
$${given}\:{f}\left({x}\right)={cos}^{\mathrm{2}} {x} \\ $$$${for}\:{x}\:\in\:\left[−\frac{\pi}{\mathrm{12}};\frac{\pi}{\mathrm{12}\:}\right]\:,\:\mid{f}'\left({x}\right)\mid\leqslant\frac{\mathrm{1}}{\mathrm{2}}. \\ $$$$\left.\mathrm{1}\right)\:{show}\:{that}\:\forall\:{x},\:{y}\:\in\:\left[−\frac{\pi}{\mathrm{12}};\frac{\pi}{\mathrm{12}}\right]\:; \\ $$$$\mid{cos}^{\mathrm{2}} {x}−{cos}^{\mathrm{2}} {y}\mid\leqslant\frac{\mathrm{1}}{\mathrm{2}}\mid{x}−{y}\mid \\ $$
Commented by ZiYangLee last updated on 16/Nov/20
$$\mathrm{By}\:\mathrm{MVT},\:\mathrm{we}\:\mathrm{have}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{f}\left({x}\right)−{f}\left({y}\right)}{{x}−{y}}={f}'\left({c}\right)\leqslant\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{where}\:{c}\in\left({x},{y}\right), \\ $$$$\mathrm{Hence}, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mid{f}\left({x}\right)−{f}\left({y}\right)\mid\leqslant\frac{\mathrm{1}}{\mathrm{2}}\mid{x}−{y}\mid \\ $$
Answered by mathmax by abdo last updated on 15/Nov/20
![∀x ∈[−(π/(12)),(π/(12))] ∣f^′ (x)∣≤(1/2) ⇒−(1/2)≤f^′ (x)≤(1/2) ⇒ −(1/2)∫_x ^y dt ≤∫_x ^y f^′ (t)dt ≤(1/2)∫_x ^y dt ⇒ −(1/2)(y−x) ≤f(y)−f(x)≤(1/2)(y−x) ⇒∣f(x)−f(y)∣≤(1/2)∣x−y∣ ⇒ ∣cos^2 x−cos^2 y∣≤((∣x−y∣)/2)](https://www.tinkutara.com/question/Q122332.png)
$$\forall\mathrm{x}\:\in\left[−\frac{\pi}{\mathrm{12}},\frac{\pi}{\mathrm{12}}\right]\:\:\mid\mathrm{f}^{'} \left(\mathrm{x}\right)\mid\leqslant\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow−\frac{\mathrm{1}}{\mathrm{2}}\leqslant\mathrm{f}^{'} \left(\mathrm{x}\right)\leqslant\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow \\ $$$$−\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{x}} ^{\mathrm{y}} \:\mathrm{dt}\:\leqslant\int_{\mathrm{x}} ^{\mathrm{y}} \:\mathrm{f}^{'} \left(\mathrm{t}\right)\mathrm{dt}\:\leqslant\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{x}} ^{\mathrm{y}} \:\mathrm{dt}\:\Rightarrow \\ $$$$−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{y}−\mathrm{x}\right)\:\leqslant\mathrm{f}\left(\mathrm{y}\right)−\mathrm{f}\left(\mathrm{x}\right)\leqslant\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{y}−\mathrm{x}\right)\:\Rightarrow\mid\mathrm{f}\left(\mathrm{x}\right)−\mathrm{f}\left(\mathrm{y}\right)\mid\leqslant\frac{\mathrm{1}}{\mathrm{2}}\mid\mathrm{x}−\mathrm{y}\mid\:\Rightarrow \\ $$$$\mid\mathrm{cos}^{\mathrm{2}} \mathrm{x}−\mathrm{cos}^{\mathrm{2}} \mathrm{y}\mid\leqslant\frac{\mid\mathrm{x}−\mathrm{y}\mid}{\mathrm{2}} \\ $$