If x is real , maximum value of frac{{3{x^2} + 9x + 17}}{{3{x^2} + 9x + 7}} is

English

Gujarati

Hindi

4-2.Quadratic Equations and Inequations

hard

If $x$ is real , the maximum value of $\frac{{3{x^2} + 9x + 17}}{{3{x^2} + 9x + 7}}$ is

A

$\frac{1}{4}$

B

$1$

C

$41$

D

$\frac{{17}}{7}$

(AIEEE-2006)

Solution

$y=\frac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7}$

$3 x^{2}(y-1)+9 x(y-1)+7 y-17=0$

$D \geq 0$ as $x$ is real

$81(y-1)^{2}-4 \times 3(y-1)(7 y-17) \geq 0$

$\Rightarrow(y-1)(y-41) \leq 0$

$\Rightarrow 1 \leq y \leq 41$

$\therefore$ Max value of $y$ is $41$

Standard 11

Mathematics

Share

Similar Questions

The number of real solutions of the equation $x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$ is

hard

(JEE MAIN-2024)

Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^2+(a+b) x+b=0$ is necessarily true?

$I$. It has at least one negative root.

$II$. It has at least one positive root.

$III$. Both its roots are real.

hard

(KVPY-2013)

The solution set of the equation $pq{x^2} – {(p + q)^2}x + {(p + q)^2} = 0$ is

medium

Let $\alpha $ and $\beta $ are roots of $5{x^2} – 3x – 1 = 0$ , then $\left[ {\left( {\alpha + \beta } \right)x – \left( {\frac{{{\alpha ^2} + {\beta ^2}}}{2}} \right){x^2} + \left( {\frac{{{\alpha ^3} + {\beta ^3}}}{3}} \right){x^3} -……} \right]$ is

normal

Solution of the equation $\sqrt {x + 3 – 4\sqrt {x – 1} } + \sqrt {x + 8 – 6\sqrt {x – 1} } = 1$ is

normal