61х - x^2=600
x^2-61-900=0
D=3721-4*900=3721-3600=121
X1,2=61+-11/2=25 или 36 (но это не точно)
Применена система двух уравнений, способ подстановки
находим производную:
![y'=(2x+22)*e^{2-x}-(x^2+22x-22)*e^{2-x}=e^{2-x}(-x^2-20x+44)=\\=-e^{2-x}(x^2+20x-44)](https://tex.z-dn.net/?f=y%27%3D%282x%2B22%29%2Ae%5E%7B2-x%7D-%28x%5E2%2B22x-22%29%2Ae%5E%7B2-x%7D%3De%5E%7B2-x%7D%28-x%5E2-20x%2B44%29%3D%5C%5C%3D-e%5E%7B2-x%7D%28x%5E2%2B20x-44%29)
приравниваем производную к нулю, находим критические точки:
![-e^{2-x}(x^2+20x-44)=0\\x^2+20x-44=0\\D=400+176=576=24^2\\x_1=\frac{-20+24}{2} =2\in[0;5]\\x_2=\frac{-20-24}{2} \notin [0;5]](https://tex.z-dn.net/?f=-e%5E%7B2-x%7D%28x%5E2%2B20x-44%29%3D0%5C%5Cx%5E2%2B20x-44%3D0%5C%5CD%3D400%2B176%3D576%3D24%5E2%5C%5Cx_1%3D%5Cfrac%7B-20%2B24%7D%7B2%7D%20%3D2%5Cin%5B0%3B5%5D%5C%5Cx_2%3D%5Cfrac%7B-20-24%7D%7B2%7D%20%5Cnotin%20%5B0%3B5%5D)
находим наибольшее значение функции на данном отрезке:
![y(5)=(25+22*5-22)*e^{-3}=113e^{-3}\\y(0)=-22e^{2}\\y(2)=(4+44-22)*e^0=26\\e\approx 2,7\Rightarrow e^{-3}=\frac{1}{2,7^3} \approx \frac{1}{20} \Rightarrow 113e^{-3}\approx \frac{113}{20} <26\\-22e^2<26\\y_{max}[0;5]=y(2)=26](https://tex.z-dn.net/?f=y%285%29%3D%2825%2B22%2A5-22%29%2Ae%5E%7B-3%7D%3D113e%5E%7B-3%7D%5C%5Cy%280%29%3D-22e%5E%7B2%7D%5C%5Cy%282%29%3D%284%2B44-22%29%2Ae%5E0%3D26%5C%5Ce%5Capprox%202%2C7%5CRightarrow%20e%5E%7B-3%7D%3D%5Cfrac%7B1%7D%7B2%2C7%5E3%7D%20%5Capprox%20%5Cfrac%7B1%7D%7B20%7D%20%5CRightarrow%20113e%5E%7B-3%7D%5Capprox%20%5Cfrac%7B113%7D%7B20%7D%20%3C26%5C%5C-22e%5E2%3C26%5C%5Cy_%7Bmax%7D%5B0%3B5%5D%3Dy%282%29%3D26)
Ответ: ![y_{max}[0;5]=y(2)=26](https://tex.z-dn.net/?f=y_%7Bmax%7D%5B0%3B5%5D%3Dy%282%29%3D26)