![(\sqrt{u})'=\frac{1}{2\sqrt{u}}\cdot u'\\\\3)\; \; y=x^3+5\sqrt{x}+7\; ,\; \; x\in [\, 4;16\, ]\; \; ,\; \; \; ODZ:\; x \geq 0\\\\y'=3x^2+5\cdot \frac{1}{2\sqrt{x}}+0=\frac{6x^2\sqrt{x}+5}{2\sqrt{x}}=0\; ,\; \; x\ne 0\; ,\\\\6x^2 \sqrt{x}+5=0\; ,\; \; \sqrt{x^{5}}=-\frac{5}{6}\ \textless \ 0\; \; (\sqrt{t} \geq 0)\; \to \; \; net\; kornej\; ,\; \to \\\\net\; kriticheskix\; tochek\; ,\; y'\ \textgreater \ 0\; \; pri\; x\ \textgreater \ 0\; .\\\\y(4)=4^3+5\sqrt4+7=81\\\\y(16)=16^3+5\sqrt{16}+7=4123\quad \to \; \; \; y_{naimen.}=81](https://tex.z-dn.net/?f=%28%5Csqrt%7Bu%7D%29%27%3D%5Cfrac%7B1%7D%7B2%5Csqrt%7Bu%7D%7D%5Ccdot+u%27%5C%5C%5C%5C3%29%5C%3B+%5C%3B++y%3Dx%5E3%2B5%5Csqrt%7Bx%7D%2B7%5C%3B+%2C%5C%3B+%5C%3B+x%5Cin+%5B%5C%2C+4%3B16%5C%2C+%5D%5C%3B+%5C%3B+%2C%5C%3B+%5C%3B+%5C%3B+ODZ%3A%5C%3B+x+%5Cgeq+0%5C%5C%5C%5Cy%27%3D3x%5E2%2B5%5Ccdot+%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%2B0%3D%5Cfrac%7B6x%5E2%5Csqrt%7Bx%7D%2B5%7D%7B2%5Csqrt%7Bx%7D%7D%3D0%5C%3B+%2C%5C%3B+%5C%3B+x%5Cne+0%5C%3B+%2C%5C%5C%5C%5C6x%5E2+%5Csqrt%7Bx%7D%2B5%3D0%5C%3B+%2C%5C%3B+%5C%3B+%5Csqrt%7Bx%5E%7B5%7D%7D%3D-%5Cfrac%7B5%7D%7B6%7D%5C+%5Ctextless+%5C+0%5C%3B+%5C%3B+%28%5Csqrt%7Bt%7D+%5Cgeq+0%29%5C%3B+%5Cto+%5C%3B+%5C%3B+net%5C%3B+kornej%5C%3B+%2C%5C%3B+%5Cto+%5C%5C%5C%5Cnet%5C%3B+kriticheskix%5C%3B+tochek%5C%3B+%2C%5C%3B+y%27%5C+%5Ctextgreater+%5C+0%5C%3B+%5C%3B+pri%5C%3B+x%5C+%5Ctextgreater+%5C+0%5C%3B+.%5C%5C%5C%5Cy%284%29%3D4%5E3%2B5%5Csqrt4%2B7%3D81%5C%5C%5C%5Cy%2816%29%3D16%5E3%2B5%5Csqrt%7B16%7D%2B7%3D4123%5Cquad+%5Cto+%5C%3B+%5C%3B+%5C%3B+y_%7Bnaimen.%7D%3D81 "(\sqrt{u})'=\frac{1}{2\sqrt{u}}\cdot u'\\\\3)\; \; y=x^3+5\sqrt{x}+7\; ,\; \; x\in [\, 4;16\, ]\; \; ,\; \; \; ODZ:\; x \geq 0\\\\y'=3x^2+5\cdot \frac{1}{2\sqrt{x}}+0=\frac{6x^2\sqrt{x}+5}{2\sqrt{x}}=0\; ,\; \; x\ne 0\; ,\\\\6x^2 \sqrt{x}+5=0\; ,\; \; \sqrt{x^{5}}=-\frac{5}{6}\ \textless \ 0\; \; (\sqrt{t} \geq 0)\; \to \; \; net\; kornej\; ,\; \to \\\\net\; kriticheskix\; tochek\; ,\; y'\ \textgreater \ 0\; \; pri\; x\ \textgreater \ 0\; .\\\\y(4)=4^3+5\sqrt4+7=81\\\\y(16)=16^3+5\sqrt{16}+7=4123\quad \to \; \; \; y_{naimen.}=81")
![4)\; \; y=(7-x)\sqrt{x+5}\; ,\; \; x\in [-4,4\, ]\; \; ,\; \; ODZ:\; \; x \geq -5\\\\y'=-1\cdot \sqrt{x+5}+(7-x)\cdot \frac{1}{2\sqrt{x+5}}\cdot 1=-\sqrt{x+5}+ \frac{7-x}{2\sqrt{x+5}}=\\\\=\frac{-2(x+5)+7-x}{2\sqrt{x+5}}=\frac{-3x-3}{2\sqrt{x+5}}=0\; \; \Rightarrow \\\\x=-1\; ,\; \; x\ne -5\; ,\; \; \quad znaki\; y'(x):\; \; (-5)+++[-1\, ]---\\\\y(-4)=(7+4)\sqrt1=3\\\\y(-1)=(7+1)\sqrt{4}=16\\\\y(4)=(7-4)\sqrt9=9\; \; \to \; \; \; y_{naibol.}=16](https://tex.z-dn.net/?f=4%29%5C%3B+%5C%3B+y%3D%287-x%29%5Csqrt%7Bx%2B5%7D%5C%3B+%2C%5C%3B+%5C%3B+x%5Cin+%5B-4%2C4%5C%2C+%5D%5C%3B+%5C%3B+%2C%5C%3B+%5C%3B+ODZ%3A%5C%3B+%5C%3B+x+%5Cgeq+-5%5C%5C%5C%5Cy%27%3D-1%5Ccdot+%5Csqrt%7Bx%2B5%7D%2B%287-x%29%5Ccdot+%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%2B5%7D%7D%5Ccdot+1%3D-%5Csqrt%7Bx%2B5%7D%2B+%5Cfrac%7B7-x%7D%7B2%5Csqrt%7Bx%2B5%7D%7D%3D%5C%5C%5C%5C%3D%5Cfrac%7B-2%28x%2B5%29%2B7-x%7D%7B2%5Csqrt%7Bx%2B5%7D%7D%3D%5Cfrac%7B-3x-3%7D%7B2%5Csqrt%7Bx%2B5%7D%7D%3D0%5C%3B+%5C%3B+%5CRightarrow+%5C%5C%5C%5Cx%3D-1%5C%3B+%2C%5C%3B+%5C%3B+x%5Cne+-5%5C%3B+%2C%5C%3B+%5C%3B+%5Cquad+znaki%5C%3B+y%27%28x%29%3A%5C%3B+%5C%3B+%28-5%29%2B%2B%2B%5B-1%5C%2C+%5D---%5C%5C%5C%5Cy%28-4%29%3D%287%2B4%29%5Csqrt1%3D3%5C%5C%5C%5Cy%28-1%29%3D%287%2B1%29%5Csqrt%7B4%7D%3D16%5C%5C%5C%5Cy%284%29%3D%287-4%29%5Csqrt9%3D9%5C%3B+%5C%3B+%5Cto+%5C%3B+%5C%3B+%5C%3B+y_%7Bnaibol.%7D%3D16 "4)\; \; y=(7-x)\sqrt{x+5}\; ,\; \; x\in [-4,4\, ]\; \; ,\; \; ODZ:\; \; x \geq -5\\\\y'=-1\cdot \sqrt{x+5}+(7-x)\cdot \frac{1}{2\sqrt{x+5}}\cdot 1=-\sqrt{x+5}+ \frac{7-x}{2\sqrt{x+5}}=\\\\=\frac{-2(x+5)+7-x}{2\sqrt{x+5}}=\frac{-3x-3}{2\sqrt{x+5}}=0\; \; \Rightarrow \\\\x=-1\; ,\; \; x\ne -5\; ,\; \; \quad znaki\; y'(x):\; \; (-5)+++[-1\, ]---\\\\y(-4)=(7+4)\sqrt1=3\\\\y(-1)=(7+1)\sqrt{4}=16\\\\y(4)=(7-4)\sqrt9=9\; \; \to \; \; \; y_{naibol.}=16")
![7)\; \; y=(x-15)\sqrt{x+12}+6\; ,\; \; x\in [-8,4\, ]\; \; ,\; \; \; ODZ:\; \; x \geq -12\\\\y'=\sqrt{x+12}+(x-15)\cdot \frac{1}{2\sqrt{x+12}}=\frac{2(x+12)+x-15}{2\sqrt{x+12}}=\frac{3x+9}{2\sqrt{x+12}}=0\\\\x\ne -12\; ,\; \; 3x+9=0\; \; ,\; \; x=-3\\\\y(-8)=-23\cdot \sqrt4=-46\\\\y(-3)=-18\cdot \sqrt9=-63\\\\y(4)=-11\cdot \sqrt{16}=-44\; \; \; \to \; \; \; y_{naimen}=-63](https://tex.z-dn.net/?f=7%29%5C%3B+%5C%3B+y%3D%28x-15%29%5Csqrt%7Bx%2B12%7D%2B6%5C%3B+%2C%5C%3B+%5C%3B+x%5Cin+%5B-8%2C4%5C%2C+%5D%5C%3B+%5C%3B+%2C%5C%3B+%5C%3B+%5C%3B+ODZ%3A%5C%3B+%5C%3B+x+%5Cgeq+-12%5C%5C%5C%5Cy%27%3D%5Csqrt%7Bx%2B12%7D%2B%28x-15%29%5Ccdot+%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%2B12%7D%7D%3D%5Cfrac%7B2%28x%2B12%29%2Bx-15%7D%7B2%5Csqrt%7Bx%2B12%7D%7D%3D%5Cfrac%7B3x%2B9%7D%7B2%5Csqrt%7Bx%2B12%7D%7D%3D0%5C%5C%5C%5Cx%5Cne+-12%5C%3B+%2C%5C%3B+%5C%3B+3x%2B9%3D0%5C%3B+%5C%3B+%2C%5C%3B+%5C%3B+x%3D-3%5C%5C%5C%5Cy%28-8%29%3D-23%5Ccdot+%5Csqrt4%3D-46%5C%5C%5C%5Cy%28-3%29%3D-18%5Ccdot+%5Csqrt9%3D-63%5C%5C%5C%5Cy%284%29%3D-11%5Ccdot+%5Csqrt%7B16%7D%3D-44%5C%3B+%5C%3B+%5C%3B+%5Cto+%5C%3B+%5C%3B+%5C%3B+y_%7Bnaimen%7D%3D-63 "7)\; \; y=(x-15)\sqrt{x+12}+6\; ,\; \; x\in [-8,4\, ]\; \; ,\; \; \; ODZ:\; \; x \geq -12\\\\y'=\sqrt{x+12}+(x-15)\cdot \frac{1}{2\sqrt{x+12}}=\frac{2(x+12)+x-15}{2\sqrt{x+12}}=\frac{3x+9}{2\sqrt{x+12}}=0\\\\x\ne -12\; ,\; \; 3x+9=0\; \; ,\; \; x=-3\\\\y(-8)=-23\cdot \sqrt4=-46\\\\y(-3)=-18\cdot \sqrt9=-63\\\\y(4)=-11\cdot \sqrt{16}=-44\; \; \; \to \; \; \; y_{naimen}=-63")
P.S. Cледим за тем, чтобы найденные критические точки входили в заданный в условии промежуток. Если какие-то точки не входят в промежуток, то и не вычисляем значение функции в них.
Вообще говоря, не обязательно определять знаки производной, чтобы знать какой экстремум будет в найденной точке, всё равно потом сравним числовые значения функции в критических точках и на концах промежутка и выберем наибольшее и наименьшее значения.