Область определения.
![y= \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1, \\
\frac{ x^{2} -3x+2}{x-1} \geq 0, \\
x-1 \neq 0, \ x \neq 1, \\
x^2-3x+2=0, \ x_1=1, \ x_2=2, \\
(x-1)^2(x-2) \geq 0, \\
x-2 \geq 0, \\
x \geq 2, \\
D_y=[2;+\infty).](https://tex.z-dn.net/?f=y%3D+%5Csqrt%5B4%5D%7B+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2%7D%7Bx-1%7D+%7D-1%2C+%5C%5C%0A+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2%7D%7Bx-1%7D+%5Cgeq+0%2C+%5C%5C%0Ax-1+%5Cneq+0%2C+%5C+x+%5Cneq+1%2C+%5C%5C%0Ax%5E2-3x%2B2%3D0%2C+%5C+x_1%3D1%2C+%5C+x_2%3D2%2C+%5C%5C%0A%28x-1%29%5E2%28x-2%29+%5Cgeq+0%2C+%5C%5C%0Ax-2+%5Cgeq+0%2C+%5C%5C%0Ax+%5Cgeq+2%2C+%5C%5C%0AD_y%3D%5B2%3B%2B%5Cinfty%29.%0A "y= \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1, \\
\frac{ x^{2} -3x+2}{x-1} \geq 0, \\
x-1 \neq 0, \ x \neq 1, \\
x^2-3x+2=0, \ x_1=1, \ x_2=2, \\
(x-1)^2(x-2) \geq 0, \\
x-2 \geq 0, \\
x \geq 2, \\
D_y=[2;+\infty).")
Область значений.
![y=\sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1, \\
y+1=\sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} } \geq 0, \\
y+1 \geq 0, \\
y \geq -1, \\
E_y=[-1;+\infty).](https://tex.z-dn.net/?f=y%3D%5Csqrt%5B4%5D%7B+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2%7D%7Bx-1%7D+%7D-1%2C+%5C%5C%0Ay%2B1%3D%5Csqrt%5B4%5D%7B+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2%7D%7Bx-1%7D+%7D+%5Cgeq+0%2C+%5C%5C%0Ay%2B1+%5Cgeq+0%2C+%5C%5C%0Ay+%5Cgeq+-1%2C+%5C%5C%0AE_y%3D%5B-1%3B%2B%5Cinfty%29. "y=\sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1, \\
y+1=\sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} } \geq 0, \\
y+1 \geq 0, \\
y \geq -1, \\
E_y=[-1;+\infty).")
Функция общего вида, т.е. ни четная ни нечетная.
Нули функции.
![x=0\notin D_y, \\ y=0, \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1=0, \\ \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }=1, \\ \frac{ x^{2} -3x+2}{x-1}=1, \\ \frac{ x^{2} -3x+2}{x-1}-1=0, \\ \frac{ x^{2} -3x+2-x+1}{x-1}=0, \\ \frac{ x^{2} -4x+3}{x-1}=0, \\ x^2-4x+3=0, \ x_1=1\notin D_y, \ x_2=3,\\ (3;0).](https://tex.z-dn.net/?f=x%3D0%5Cnotin+D_y%2C+%5C%5C+y%3D0%2C+%5Csqrt%5B4%5D%7B+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2%7D%7Bx-1%7D+%7D-1%3D0%2C+%5C%5C+%5Csqrt%5B4%5D%7B+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2%7D%7Bx-1%7D+%7D%3D1%2C+%5C%5C+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2%7D%7Bx-1%7D%3D1%2C+%5C%5C+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2%7D%7Bx-1%7D-1%3D0%2C+%5C%5C+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2-x%2B1%7D%7Bx-1%7D%3D0%2C+%5C%5C+%5Cfrac%7B+x%5E%7B2%7D+-4x%2B3%7D%7Bx-1%7D%3D0%2C+%5C%5C+x%5E2-4x%2B3%3D0%2C+%5C+x_1%3D1%5Cnotin+D_y%2C+%5C+x_2%3D3%2C%5C%5C+%283%3B0%29. "x=0\notin D_y, \\ y=0, \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1=0, \\ \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }=1, \\ \frac{ x^{2} -3x+2}{x-1}=1, \\ \frac{ x^{2} -3x+2}{x-1}-1=0, \\ \frac{ x^{2} -3x+2-x+1}{x-1}=0, \\ \frac{ x^{2} -4x+3}{x-1}=0, \\ x^2-4x+3=0, \ x_1=1\notin D_y, \ x_2=3,\\ (3;0).")
Промежутки знакопостоянства.
3, x\in(3;+\infty), y>0." alt="y\gtrless0, \\ \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1\gtrless0, \\ \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }\gtrless1, \\ \frac{ x^{2} -3x+2}{x-1}\gtrless1, \\ \frac{ x^{2} -3x+2}{x-1}-1\gtrless0, \\ \frac{ x^{2} -4x+3}{x-1}\gtrless0, \\ \frac{(x-1)(x-3)}{x-1}\gtrless0, \\ (x-1)^2(x-3)\gtrless0, \\ x-3\gtrless0, \\ x\gtrless3, \\ 23, x\in(3;+\infty), y>0." align="absmiddle" class="latex-formula">
Производные функции.
![y=\sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1, \\ y'=(( \frac{ x^{2} -3x+2}{x-1})^{ \frac{1}{4}}-1)' =(( \frac{ x^{2} -3x+2}{x-1})^{ \frac{1}{4}})'= \\ =\frac{1}{4} ( \frac{ x^{2} -3x+2}{x-1})^{- \frac{3}{4}}\cdot \frac{(x^{2} -3x+2)'(x-1)-(x^{2} -3x+2)(x-1)'}{(x-1)^2} = \\ =\frac{1}{4} ( \frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}}\cdot \frac{(2x-3)(x-1)-(x^2 -3x+2)}{(x-1)^2} = \\ =\frac{1}{4} ( \frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}}\cdot \frac{2x^2-5x+3-x^2 +3x-2}{(x-1)^2} =\\ =](https://tex.z-dn.net/?f=y%3D%5Csqrt%5B4%5D%7B+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2%7D%7Bx-1%7D+%7D-1%2C+%5C%5C+y%27%3D%28%28+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2%7D%7Bx-1%7D%29%5E%7B+%5Cfrac%7B1%7D%7B4%7D%7D-1%29%27+%3D%28%28+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2%7D%7Bx-1%7D%29%5E%7B+%5Cfrac%7B1%7D%7B4%7D%7D%29%27%3D+%5C%5C+%3D%5Cfrac%7B1%7D%7B4%7D+%28+%5Cfrac%7B+x%5E%7B2%7D+-3x%2B2%7D%7Bx-1%7D%29%5E%7B-+%5Cfrac%7B3%7D%7B4%7D%7D%5Ccdot+%5Cfrac%7B%28x%5E%7B2%7D+-3x%2B2%29%27%28x-1%29-%28x%5E%7B2%7D+-3x%2B2%29%28x-1%29%27%7D%7B%28x-1%29%5E2%7D+%3D+%5C%5C+%3D%5Cfrac%7B1%7D%7B4%7D+%28+%5Cfrac%7Bx-1%7D%7B+x%5E%7B2%7D+-3x%2B2%7D%29%5E%7B+%5Cfrac%7B3%7D%7B4%7D%7D%5Ccdot+%5Cfrac%7B%282x-3%29%28x-1%29-%28x%5E2+-3x%2B2%29%7D%7B%28x-1%29%5E2%7D+%3D+%5C%5C+%3D%5Cfrac%7B1%7D%7B4%7D+%28+%5Cfrac%7Bx-1%7D%7B+x%5E%7B2%7D+-3x%2B2%7D%29%5E%7B+%5Cfrac%7B3%7D%7B4%7D%7D%5Ccdot+%5Cfrac%7B2x%5E2-5x%2B3-x%5E2+%2B3x-2%7D%7B%28x-1%29%5E2%7D+%3D%5C%5C+%3D "y=\sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1, \\ y'=(( \frac{ x^{2} -3x+2}{x-1})^{ \frac{1}{4}}-1)' =(( \frac{ x^{2} -3x+2}{x-1})^{ \frac{1}{4}})'= \\ =\frac{1}{4} ( \frac{ x^{2} -3x+2}{x-1})^{- \frac{3}{4}}\cdot \frac{(x^{2} -3x+2)'(x-1)-(x^{2} -3x+2)(x-1)'}{(x-1)^2} = \\ =\frac{1}{4} ( \frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}}\cdot \frac{(2x-3)(x-1)-(x^2 -3x+2)}{(x-1)^2} = \\ =\frac{1}{4} ( \frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}}\cdot \frac{2x^2-5x+3-x^2 +3x-2}{(x-1)^2} =\\ =")
Критические точки.
![y'=0, \frac{1}{4} ( \frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}}=0, \\ \frac{x-1}{ x^{2} -3x+2}=0, \\ x^{2} -3x+2 \neq0,\\ (x-1)(x-2) \neq 0, \\ x \neq 1\notin D_y, \\ x \neq 2; \\ x=2, y= \sqrt[4]{ \frac{ 2^{2} -3\cdot2+2}{2-1} }-1=\sqrt[4]{4 -6+2}-1=\sqrt[4]{0}-1=-1;\\ (2;-1).](https://tex.z-dn.net/?f=y%27%3D0%2C+%5Cfrac%7B1%7D%7B4%7D+%28+%5Cfrac%7Bx-1%7D%7B+x%5E%7B2%7D+-3x%2B2%7D%29%5E%7B+%5Cfrac%7B3%7D%7B4%7D%7D%3D0%2C+%5C%5C+%5Cfrac%7Bx-1%7D%7B+x%5E%7B2%7D+-3x%2B2%7D%3D0%2C+%5C%5C+x%5E%7B2%7D+-3x%2B2+%5Cneq0%2C%5C%5C+%28x-1%29%28x-2%29+%5Cneq+0%2C+%5C%5C+x+%5Cneq+1%5Cnotin+D_y%2C+%5C%5C+x+%5Cneq+2%3B+%5C%5C+x%3D2%2C+y%3D+%5Csqrt%5B4%5D%7B+%5Cfrac%7B+2%5E%7B2%7D+-3%5Ccdot2%2B2%7D%7B2-1%7D+%7D-1%3D%5Csqrt%5B4%5D%7B4+-6%2B2%7D-1%3D%5Csqrt%5B4%5D%7B0%7D-1%3D-1%3B%5C%5C+%282%3B-1%29. "y'=0, \frac{1}{4} ( \frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}}=0, \\ \frac{x-1}{ x^{2} -3x+2}=0, \\ x^{2} -3x+2 \neq0,\\ (x-1)(x-2) \neq 0, \\ x \neq 1\notin D_y, \\ x \neq 2; \\ x=2, y= \sqrt[4]{ \frac{ 2^{2} -3\cdot2+2}{2-1} }-1=\sqrt[4]{4 -6+2}-1=\sqrt[4]{0}-1=-1;\\ (2;-1).")
Промежутки монотонности.
<img src="https://tex.z-dn.net/?f=y%27%5Cgtrless0%2C+%5C+%5Cfrac%7B1%7D%7B4%7D+%28%5Cfrac%7Bx-1%7D%7B+x%5E%7B2%7D+-3x%2B2%7D%29%5E%7B+%5Cfrac%7B3%7D%7B4