Question Number 213290 by issac last updated on 02/Nov/24
$$\mathrm{for}\:\mathrm{every}\:\mathrm{real}\:\mathrm{set}\:\mathbb{R}\:,\:{f}\in\mathbb{R} \\ $$$$\mathrm{and}\:{f}\:\mathrm{is}\:\mathrm{Smooth}\:\mathrm{function}.\:\mathrm{and}\:{f}\:\mathrm{is}\:{f}\in\mathcal{C}^{\mathrm{2}} \\ $$$$\forall_{{x}} \:{f}^{\left(\mathrm{1}\right)} \left({x}\right)>\mathrm{0}\:,\:{f}^{\left(\mathrm{2}\right)} \left({x}\right)<\mathrm{0} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mid\int_{\mathrm{0}} ^{\:{t}} \:\mathrm{cos}\left({f}\left({x}\right)\right)\mathrm{d}{x}\mid\leq\frac{\mathrm{2}}{{f}^{\left(\mathrm{1}\right)} \left({t}\right)} \\ $$$${t}\in\mathbb{R} \\ $$
Answered by MrGaster last updated on 02/Nov/24
![∣∫_0 ^t cos(f(x))dx∣ =∣[((sin(f(x)))/(f′(x)))]_0 ^t −∫_0 ^t ((sin(f(x))∫^(′′) (x))/((f′(x))^2 ))dx∣ ≤∣((∣sin(f(t))∣)/(f′(t)))+((∣sin(f(0))∣)/(f′(0)))+∫_0 ^t ((∣sin(f(x))f′′(x))/((f′(x))^2 ))dx ≤((∣sin(f(t))∣)/(f′(t)))+((∣sin(f(0))∣)/(f′(0)))+∫_0 ^t ((∣sin(f(x))∣∣f′′(x)∣)/((f′(x))^2 ))dx ≤(1/(f′(t)))+(1/(f′(0)))+∫_0 ^t ((∣f^(′′) (x)∣)/((f′(x))^2 ))dx ≤(1/(f′(t)))+(1/(f′(0)))+∫_0 ^t ((−f′′(x))/((f′(x))^2 ))dx =(1/(f′(t)))+(1/(f′(0)))+[−(1/(f′(x)))]_0 ^t =(1/(f′(t)))+(1/(f′(0)))−(1/(f′(t)))+(1/(f′(0))) =(2/(f′(0))) ≤(2/(f′(t)))](https://www.tinkutara.com/question/Q213293.png)
$$\mid\int_{\mathrm{0}} ^{{t}} \mathrm{cos}\left({f}\left({x}\right)\right){dx}\mid \\ $$$$=\mid\left[\frac{\mathrm{sin}\left({f}\left({x}\right)\right)}{{f}'\left({x}\right)}\right]_{\mathrm{0}} ^{{t}} −\int_{\mathrm{0}} ^{{t}} \frac{\mathrm{sin}\left({f}\left({x}\right)\right)\int^{''} \left({x}\right)}{\left({f}'\left({x}\right)\right)^{\mathrm{2}} }{dx}\mid \\ $$$$\leq\mid\frac{\mid\mathrm{sin}\left({f}\left({t}\right)\right)\mid}{{f}'\left({t}\right)}+\frac{\mid\mathrm{sin}\left({f}\left(\mathrm{0}\right)\right)\mid}{{f}'\left(\mathrm{0}\right)}+\int_{\mathrm{0}} ^{{t}} \frac{\mid\mathrm{sin}\left({f}\left({x}\right)\right){f}''\left({x}\right)}{\left({f}'\left({x}\right)\right)^{\mathrm{2}} }{dx} \\ $$$$\leq\frac{\mid\mathrm{sin}\left({f}\left({t}\right)\right)\mid}{{f}'\left({t}\right)}+\frac{\mid\mathrm{sin}\left({f}\left(\mathrm{0}\right)\right)\mid}{{f}'\left(\mathrm{0}\right)}+\int_{\mathrm{0}} ^{{t}} \frac{\mid\mathrm{sin}\left({f}\left({x}\right)\right)\mid\mid{f}''\left({x}\right)\mid}{\left({f}'\left({x}\right)\right)^{\mathrm{2}} }{dx} \\ $$$$\leq\frac{\mathrm{1}}{{f}'\left({t}\right)}+\frac{\mathrm{1}}{{f}'\left(\mathrm{0}\right)}+\int_{\mathrm{0}} ^{{t}} \frac{\mid{f}^{''} \left({x}\right)\mid}{\left({f}'\left({x}\right)\right)^{\mathrm{2}} }{dx} \\ $$$$\leq\frac{\mathrm{1}}{{f}'\left({t}\right)}+\frac{\mathrm{1}}{{f}'\left(\mathrm{0}\right)}+\int_{\mathrm{0}} ^{{t}} \frac{−{f}''\left({x}\right)}{\left({f}'\left({x}\right)\right)^{\mathrm{2}} }{dx} \\ $$$$=\frac{\mathrm{1}}{{f}'\left({t}\right)}+\frac{\mathrm{1}}{{f}'\left(\mathrm{0}\right)}+\left[−\frac{\mathrm{1}}{{f}'\left({x}\right)}\right]_{\mathrm{0}} ^{{t}} \\ $$$$=\frac{\mathrm{1}}{{f}'\left({t}\right)}+\frac{\mathrm{1}}{{f}'\left(\mathrm{0}\right)}−\frac{\mathrm{1}}{{f}'\left({t}\right)}+\frac{\mathrm{1}}{{f}'\left(\mathrm{0}\right)} \\ $$$$=\frac{\mathrm{2}}{{f}'\left(\mathrm{0}\right)} \\ $$$$\leq\frac{\mathrm{2}}{{f}'\left({t}\right)} \\ $$