Question

Постройте график функции $y= \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1$

Answer 1

Область определения.
$y= \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1, \\ \frac{ x^{2} -3x+2}{x-1} \geq 0, \\ x-1 \neq 0, \ x \neq 1, \\ x^2-3x+2=0, \ x_1=1, \ x_2=2, \\ (x-1)^2(x-2) \geq 0, \\ x-2 \geq 0, \\ x \geq 2, \\ D_y=[2;+\infty).$

Область значений.
$y=\sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1, \\ y+1=\sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} } \geq 0, \\ y+1 \geq 0, \\ y \geq -1, \\ E_y=[-1;+\infty).$

Функция общего вида, т.е. ни четная ни нечетная.

Нули функции.
$x=0\notin D_y, \\ y=0, \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1=0, \\ \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }=1, \\ \frac{ x^{2} -3x+2}{x-1}=1, \\ \frac{ x^{2} -3x+2}{x-1}-1=0, \\ \frac{ x^{2} -3x+2-x+1}{x-1}=0, \\ \frac{ x^{2} -4x+3}{x-1}=0, \\ x^2-4x+3=0, \ x_1=1\notin D_y, \ x_2=3,\\ (3;0).$

Промежутки знакопостоянства.
$y\gtrless0, \\ \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1\gtrless0, \\ \sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }\gtrless1, \\ \frac{ x^{2} -3x+2}{x-1}\gtrless1, \\ \frac{ x^{2} -3x+2}{x-1}-1\gtrless0, \\ \frac{ x^{2} -4x+3}{x-1}\gtrless0, \\ \frac{(x-1)(x-3)}{x-1}\gtrless0, \\ (x-1)^2(x-3)\gtrless0, \\ x-3\gtrless0, \\ x\gtrless3, \\ 2<x<3, x\in(2;3), y<0, \\ x>3, x\in(3;+\infty), y>0.$

Производные функции.
$y=\sqrt[4]{ \frac{ x^{2} -3x+2}{x-1} }-1, \\ y'=(( \frac{ x^{2} -3x+2}{x-1})^{ \frac{1}{4}}-1)' =(( \frac{ x^{2} -3x+2}{x-1})^{ \frac{1}{4}})'= \\ =\frac{1}{4} ( \frac{ x^{2} -3x+2}{x-1})^{- \frac{3}{4}}\cdot \frac{(x^{2} -3x+2)'(x-1)-(x^{2} -3x+2)(x-1)'}{(x-1)^2} = \\ =\frac{1}{4} ( \frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}}\cdot \frac{(2x-3)(x-1)-(x^2 -3x+2)}{(x-1)^2} = \\ =\frac{1}{4} ( \frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}}\cdot \frac{2x^2-5x+3-x^2 +3x-2}{(x-1)^2} =\\ =$ $\frac{1}{4} ( \frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}}\cdot \frac{x^2-2x+1}{(x-1)^2}=\frac{1}{4} (\frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}}\cdot \frac{(x-1)^2}{(x-1)^2}=\frac{1}{4} (\frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}.$
$y''=(\frac{1}{4} (\frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}})'=\frac{1}{4}\cdot \frac{3}{4} (\frac{x-1}{ x^{2} -3x+2})^{-\frac{1}{4}}(\frac{x-1}{ x^{2} -3x+2})'= \\ =\frac{3}{16} (\frac{x^{2} -3x+2}{ x-1})^{\frac{1}{4}}\frac{(x-1)'(x^{2} -3x+2)-(x-1)(x^{2} -3x+2)'}{(x^{2} -3x+2)^2}= \\ =\frac{3}{16} (\frac{x^{2} -3x+2}{ x-1})^{\frac{1}{4}}\frac{x^{2} -3x+2-(x-1)(2x-3)}{((x-1)(x-2))^2}= \\ =\frac{3}{16} (\frac{x^{2} -3x+2}{ x-1})^{\frac{1}{4}}\frac{x^{2} -3x+2-2x^2+5x-3}{(x-1)^2(x-2)^2}= \\ =$ $=\frac{3}{16} (\frac{x^{2} -3x+2}{ x-1})^{\frac{1}{4}}\frac{-2x^2+2x-1}{(x-1)^2(x-2)^2}=\frac{3}{16} (\frac{x^{2} -3x+2}{ x-1})^{\frac{1}{4}}\frac{-(x-1)^2}{(x-1)^2(x-2)^2}=\\ =-\frac{3}{16} (\frac{x^{2} -3x+2}{ x-1})^{\frac{1}{4}}\frac{1}{(x-2)^2}$

Критические точки.
$y'=0, \frac{1}{4} ( \frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}}=0, \\ \frac{x-1}{ x^{2} -3x+2}=0, \\ x^{2} -3x+2 \neq0,\\ (x-1)(x-2) \neq 0, \\ x \neq 1\notin D_y, \\ x \neq 2; \\ x=2, y= \sqrt[4]{ \frac{ 2^{2} -3\cdot2+2}{2-1} }-1=\sqrt[4]{4 -6+2}-1=\sqrt[4]{0}-1=-1;\\ (2;-1).$

Промежутки монотонности.
$y'\gtrless0, \ \frac{1}{4} (\frac{x-1}{ x^{2} -3x+2})^{ \frac{3}{4}}\gtrless0, \\ \frac{1}{4} \sqrt[4]{ (\frac{x-1}{ x^{2} -3x+2})^3} >0, \\ x\in D_y,\ y'>0,\ y\nearrow.$

Промежутки выпуклости-вогнутости.
$y''\gtrless0, \ -\frac{3}{16} (\frac{x^{2} -3x+2}{ x-1})^{\frac{1}{4}}\frac{1}{(x-2)^2}\gtrless0, \\ (\frac{x^{2} -3x+2}{ x-1})^{\frac{1}{4}} \geq 0, \frac{1}{(x-2)^2}>0,\\ -\frac{3}{16} (\frac{x^{2} -3x+2}{ x-1})^{\frac{1}{4}}\frac{1}{(x-2)^2}<0, \\ x\in D_y, y''<0, y\smallfrown.$