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Question Number 29778 by puneet1789 last updated on 12/Feb/18

Answered by Giannibo last updated on 12/Feb/18

$$$$$$$$$$\mathrm{f}\left(\mathrm{x}\right)+{\mathrm{f}}^{\prime }\left(\mathrm{x}\right)⩽1$$$${\mathrm{e}}^{\mathrm{x}}\mathrm{f}\left(\mathrm{x}\right)+{\mathrm{e}}^{\mathrm{x}}{\mathrm{f}}^{\prime }\left(\mathrm{x}\right)⩽{\mathrm{e}}^{\mathrm{x}}$$$${\int }_0^1{\left({\mathrm{e}}^{\mathrm{x}}\mathrm{f}\left(\mathrm{x}\right)\right)}^{\prime }\mathrm{dx}⩽{\int }_0^1{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$$$${\left[{\mathrm{e}}^{\mathrm{x}}\mathrm{f}\left(\mathrm{x}\right)\right]}_0^1⩽{\left[{\mathrm{e}}^{\mathrm{x}}\right]}_0^1$$$$\mathrm{ef}\left(1\right)-\mathrm{f}\left(0\right)⩽\mathrm{e}-1$$$$\mathrm{f}\left(1\right)⩽\frac{\mathrm{e}-1}{\mathrm{e}}$$

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