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Question Number 213290 by issac last updated on 02/Nov/24

$$\mathrm{for}\mathrm{every}\mathrm{real}\mathrm{set}\mathbb{R},f\in \mathbb{R}$$$$\mathrm{and}f\mathrm{is}\mathrm{Smooth}\mathrm{function}.\mathrm{and}f\mathrm{is}f\in {\mathcal{C}}^2$$$${\mathrm{\forall }}_x{f}^{\left(1\right)}\left(x\right)>0,f^{\left(2\right)}\left(x\right)<0$$$$\mathrm{then}\mathrm{prove}\mid {\int }_0^t\cos \left(f\left(x\right)\right)\mathrm{d}x\mid \le \frac{2}{f^{\left(1\right)}\left(t\right)}$$$$t\in \mathbb{R}$$

Answered by MrGaster last updated on 02/Nov/24

$$\mid {\int }_0^t\cos \left(f\left(x\right)\right)dx\mid$$$$=\mid {\left[\frac{\sin \left(f\left(x\right)\right)}{f^{\prime }\left(x\right)}\right]}_0^t-{\int }_0^t\frac{\sin \left(f\left(x\right)\right){\int }^{{}^{″}}\left(x\right)}{{\left(f^{\prime }\left(x\right)\right)}^2}dx\mid$$$$\le \mid \frac{\mid \sin \left(f\left(t\right)\right)\mid }{f^{\prime }\left(t\right)}+\frac{\mid \sin \left(f\left(0\right)\right)\mid }{f^{\prime }\left(0\right)}+{\int }_0^t\frac{\mid \sin \left(f\left(x\right)\right)f^{″}\left(x\right)}{{\left(f^{\prime }\left(x\right)\right)}^2}dx$$$$\le \frac{\mid \sin \left(f\left(t\right)\right)\mid }{f^{\prime }\left(t\right)}+\frac{\mid \sin \left(f\left(0\right)\right)\mid }{f^{\prime }\left(0\right)}+{\int }_0^t\frac{\mid \sin \left(f\left(x\right)\right)\mid \mid f^{″}\left(x\right)\mid }{{\left(f^{\prime }\left(x\right)\right)}^2}dx$$$$\le \frac{1}{f^{\prime }\left(t\right)}+\frac{1}{f^{\prime }\left(0\right)}+{\int }_0^t\frac{\mid f^{{}^{″}}\left(x\right)\mid }{{\left(f^{\prime }\left(x\right)\right)}^2}dx$$$$\le \frac{1}{f^{\prime }\left(t\right)}+\frac{1}{f^{\prime }\left(0\right)}+{\int }_0^t\frac{-f^{″}\left(x\right)}{{\left(f^{\prime }\left(x\right)\right)}^2}dx$$$$=\frac{1}{f^{\prime }\left(t\right)}+\frac{1}{f^{\prime }\left(0\right)}+{[-\frac{1}{f^{\prime }\left(x\right)}]}_0^t$$$$=\frac{1}{f^{\prime }\left(t\right)}+\frac{1}{f^{\prime }\left(0\right)}-\frac{1}{f^{\prime }\left(t\right)}+\frac{1}{f^{\prime }\left(0\right)}$$$$=\frac{2}{f^{\prime }\left(0\right)}$$$$\le \frac{2}{f^{\prime }\left(t\right)}$$

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