﻿ Find the minimum and maximum of the function y=\dfrac{x^2+x^2+2}{2x^2-x+1}

#### Question

Find the minimum and maximum of the function

y=\dfrac{x^2+x^2+2}{2x^2-x+1}

Collected in the board: Function

Steven Zheng posted 4 months ago

Rearrange the function

y(2x^2-x+1) = x^2+x^2+2

(1-2y)x^2+(1+y)x+2-y = 0

For the quadratic equation to have real solutions,

the discriminant must be greater than or equal to 0, that is,

(1+y)^2-4(1-2y)(2-y)\geq 0

Simplifying the inequality gives

7y^2-22y+7\leq 0
(1)

Divide the equation by 7 to make the coefficient of leading term 1

y^2-\dfrac{22}{7}y+1\leq 0
(2)

On the other hand,

suppose a,b are the minimum and maximum of the function y, that is

a< y < b

Then

(y-a)(y-b)<0

Expand the equation

y^2-(a+b)+ab<0
(3)

Comparing coefficients of each terms of (3) with those of (1) results in

a+b = \dfrac{22}{7}

ab = 1

Solving the system of equations for a,b yields

y_{min} = \dfrac{11-6\sqrt{2} }{7}

and

y_{man} = \dfrac{11+6\sqrt{2} }{7}

Steven Zheng posted 4 months ago

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