Найдите наименьшее и наибольшее значения функции y=-√x-1+3 на отрезке [0; 5]
Производная не существует при х=1, значит х=1 - критическая точка.
у' = -1/2√(х-1);
у' = 0;
√х-1 = 0;
х-1 = 0;
х = 1 - стац.точка.
у(0) = не существует.
у(1) = 3;
у(5) = 1.
ответ :
у(наиб) = у(1) = 3;
у(наим) = у(5) = 1.
![y=-\sqrt{x-1}+3\; \; ,\; \; x\in [\, 0;5\, ]\\\\ODZ:\; x-1\geq 0\; \; \to \; \; x\geq 1\\\\y'=-\frac{1}{2\sqrt{x-1}}=0\; \; \to \; \; x\in \varnothing \; ,\; \; x\ne 1\\\\](https://tex.z-dn.net/?f=y%3D-%5Csqrt%7Bx-1%7D%2B3%5C%3B%20%5C%3B%20%2C%5C%3B%20%5C%3B%20x%5Cin%20%5B%5C%2C%200%3B5%5C%2C%20%5D%5C%5C%5C%5CODZ%3A%5C%3B%20x-1%5Cgeq%200%5C%3B%20%5C%3B%20%5Cto%20%5C%3B%20%5C%3B%20x%5Cgeq%201%5C%5C%5C%5Cy%27%3D-%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx-1%7D%7D%3D0%5C%3B%20%5C%3B%20%5Cto%20%5C%3B%20%5C%3B%20x%5Cin%20%5Cvarnothing%20%5C%3B%20%2C%5C%3B%20%5C%3B%20x%5Cne%201%5C%5C%5C%5C "y=-\sqrt{x-1}+3\; \; ,\; \; x\in [\, 0;5\, ]\\\\ODZ:\; x-1\geq 0\; \; \to \; \; x\geq 1\\\\y'=-\frac{1}{2\sqrt{x-1}}=0\; \; \to \; \; x\in \varnothing \; ,\; \; x\ne 1\\\\")
![1\in [\, 0;5\, ]\\\\y(1)=-\sqrt{0}+3=3\\\\y(0)\; ne\; syshestvyet\; ,\; 0\notin ODZ\\\\y(5)=-2+3=1\\\\y_{naibolshee}=y(1)=3\\\\y_{naimenshee}=y(5)=1](https://tex.z-dn.net/?f=1%5Cin%20%5B%5C%2C%200%3B5%5C%2C%20%5D%5C%5C%5C%5Cy%281%29%3D-%5Csqrt%7B0%7D%2B3%3D3%5C%5C%5C%5Cy%280%29%5C%3B%20ne%5C%3B%20syshestvyet%5C%3B%20%2C%5C%3B%200%5Cnotin%20ODZ%5C%5C%5C%5Cy%285%29%3D-2%2B3%3D1%5C%5C%5C%5Cy_%7Bnaibolshee%7D%3Dy%281%29%3D3%5C%5C%5C%5Cy_%7Bnaimenshee%7D%3Dy%285%29%3D1 "1\in [\, 0;5\, ]\\\\y(1)=-\sqrt{0}+3=3\\\\y(0)\; ne\; syshestvyet\; ,\; 0\notin ODZ\\\\y(5)=-2+3=1\\\\y_{naibolshee}=y(1)=3\\\\y_{naimenshee}=y(5)=1")