Найдите наименьшее значение функции y=(x−11)e^x−10 на отрезке [9;11].

[latex]\displaystyle f(x)=(x-11)e^x-10=xe^x-11e^x-10;[/latex] [latex]\displaystyle f'(x)=\frac{df(x)}{dx}=\frac{d}{dx}\left(xe^x-11e^x-10\right)=\frac{d}{dx}\left(xe^x\right)-\frac{d}{dx}\left(11e^x\right)-\frac{d}{dx}\left(10\right)=\frac{xd\left(e^x\right)+e^xdx}{dx}-11\frac{d}{dx}\left(e^x\right)-0=\frac{xe^xdx}{dx}+\frac{e^xdx}{dx}-11e^x=xe^x+e^x-11e^x=xe^x-10e^x=e^x(x-10);[/latex] [latex]\displaystyle 0=f'(x)=e^x(x-10);[/latex] [latex]\displaystyle \forall x\in\mathbb{R}: e^x\neq 0 \implies \Big(e^x\left(x-10\right)=0 \iff x-10=0\Big);[/latex] [latex]\displaystyle x-10=0\implies x=10;[/latex] [latex]\displaystyle f(9)=(9-11)e^9-10=-2e^9-10;[/latex] [latex]\displaystyle f(10)=(10-11)e^{10}-10=-e^{10}-10;[/latex] [latex]\displaystyle f(11)=(11-11)e^{11}-10=-10;[/latex] [latex]\displaystyle f(9)\stackrel{?}{=}f(10);[/latex] [latex]\displaystyle -2e^9-10\stackrel{?}{=}-e^{10}-10;[/latex] [latex]\displaystyle -2e^9\stackrel{?}{=}-e^{10};[/latex] [latex]\displaystyle 2e^9\stackrel{?}{=}e^{10};[/latex] [latex]\displaystyle ln(2e^9)\stackrel{?}{=}ln(e^{10});[/latex] [latex]\displaystyle ln(2)+ln(e^9)\stackrel{?}{=}10;[/latex] [latex]\displaystyle ln(2)+9\stackrel{?}{=}10;[/latex] [latex]\displaystyle ln(2)\stackrel{?}{=}1;[/latex] [latex]\displaystyle e^{ln(2)}\stackrel{?}{=}e^1;[/latex] [latex]\displaystyle 2\stackrel{?}{=}e;[/latex] [latex]\displaystyle 2-e\implies -2e^9>-e^{10}\implies f(9)>f(10);[/latex] [latex]\displaystyle f(10)\stackrel{?}{=}f(11);[/latex] [latex]\displaystyle -e^{10}-10\stackrel{?}{=}-10;[/latex] [latex]\displaystyle -e^{10}-10+10\stackrel{?}{=}0;[/latex] [latex]\displaystyle -e^{10}<0 \implies f(10)f(10)\land f(10)