What is the minimum value of frac{1 - x + x^2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $y = \frac{1 - x + x^2}{1 + x + x^2}$.We can rewrite this as $y = \frac{(1 + x + x^2) - 2x}{1 + x + x^2} = 1 - \frac{2x}{1 + x + x^2}$.For $x

Vedclass · Accepted Answer

$1/3$

Vedclass · Answer

(B) Let $y = \frac{1 - x + x^2}{1 + x + x^2}$.We can rewrite this as $y = \frac{(1 + x + x^2) - 2x}{1 + x + x^2} = 1 - \frac{2x}{1 + x + x^2}$.For $x = 0$,$y = 1$.For $x 
eq 0$,$y = 1 - \frac{2}{\frac{1}{x} + 1 + x}$.By $AM$-$GM$ inequality,for $x > 0$,$\frac{1}{x} + x \geq 2$,so $\frac{1}{x} + 1 + x \geq 3$.Thus,$\frac{2}{\frac{1}{x} + 1 + x} \leq \frac{2}{3}$.Therefore,$y = 1 - \frac{2}{\frac{1}{x} + 1 + x} \geq 1 - \frac{2}{3} = \frac{1}{3}$.For $x  0$. Then $y = \frac{1 + t + t^2}{1 - t + t^2} = \frac{1}{\frac{1 - t + t^2}{1 + t + t^2}}$.Since the minimum value of the expression is $1/3$,the maximum value is $3$. Thus $y \geq 1/3$ for all $x$.The minimum value is $1/3$.

What is the minimum value of $\frac{1 - x + x^2}{1 + x + x^2}$?

Similar Questions

If a polynomial $P(x)$ of degree $4$ is given by $P(x) = 2x^4 + ax^3 + bx^2 + cx + d$ such that $P(1) = 4, P(2) = 7, P(3) = 12$,and $P(4) = 19$,then find the value of $P(5)$.

Let $\alpha, \beta$ be the roots of the equation $x^2 - |a|x - |b| = 0$ such that $|\alpha| < |\beta|$. If $|a| < \beta - 1$,then the positive root of $\log_{|\alpha|} \left( \frac{x^2}{\beta^2} \right) - 1 = 0$ is

Let $S = \{x \in [-6, 3] \setminus \{-2, 2\} : \frac{|x+3|-1}{|x|-2} \geq 0\}$ and $T = \{x \in \mathbb{Z} : x^2 - 7|x| + 9 \leq 0\}$. Then the number of elements in $S \cap T$ is $....$

For what condition will the expression $a^2x^2 + bx + 1$ be positive for all $x \in R$?

Let $a, b, c, d$ be real numbers between $-5$ and $5$ such that $|a|=\sqrt{4-\sqrt{5-a}}$,$|b|=\sqrt{4+\sqrt{5-b}}$,$|c|=\sqrt{4-\sqrt{5+c}}$,and $|d|=\sqrt{4+\sqrt{5+d}}$. Then,the product $abcd$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module