(D) Given $ax^2 + bx + c < 0$ for all $x \in R$. This implies $a < 0$ and $D = b^2 - 4ac < 0$. The extreme value of $ax^2 + bx + c$ occurs at $x = -\f

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

(D) Given $ax^2 + bx + c The extreme value of $ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$. The extreme value of $cx^2 + ax + b$ occurs at $x = -\frac{a}{2c}$. Since these points are the same,$-\frac{b}{2a} = -\frac{a}{2c}$,which implies $a^2 = bc$. Since $a The expression $cx^2 + ax + b$ has a maximum value because $c The maximum value is given by $-\frac{D'}{4c} = -\frac{a^2 - 4bc}{4c} = -\frac{bc - 4bc}{4c} = -\frac{-3bc}{4c} = \frac{3b}{4}$. Thus,the maximum value is $\frac{3b}{4}$.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Quadratic expressions and Position of roots

Difficult

If $ax^2 + bx + c < 0$ for all $x \in R$ and the expressions $cx^2 + ax + b$ and $ax^2 + bx + c$ have their extreme values at the same point $x$,then for the expression $cx^2 + ax + b$:

AP EAMCET 2025 All AP EAMCET

A
Minimum value $= \frac{4b}{3}$
B
Maximum value $= \frac{4a}{3}$
C
Minimum value $= \frac{3a}{4}$
D
Maximum value $= \frac{3b}{4}$

Explore More

Browse All

Question Bank

All Subjects

Class 11 Questions

Class 11

Mathematics Questions

Chapter

4-2.Quadratic Equations and Inequations

Subtopic

Quadratic expressions and Position of roots

Previous Year

AP EAMCET 2025 Paper All AP EAMCET years →

For what set of values of $K$ do both roots of the equation $4x^2 - 20Kx + (25K^2 + 15K - 66) = 0$ lie below $2$?

Difficult

View Solution

The number of integral values of $m$ for which the quadratic expression $(1 + 2m)x^2 - 2(1 + 3m)x + 4(1 + m)$ is always positive for all $x \in R$ is:

DifficultJEE MAIN 2019

View Solution

If $b > a$,then the equation $(x - a)(x - b) = 1$ has

MediumIIT 2000

View Solution

The real number $k$ for which the equation $2x^2 + 3x + k = 0$ has two distinct real roots in the interval $[0, 1]$.

DifficultJEE MAIN 2013

View Solution

Match the following: Consider the equation $x^2 + 2(a - 1)x + a + 5 = 0$. Match the real values of $a$ with the conditions on the roots of the given equation.

Column-$I$ Column-$II$

$A$. Imaginary roots $P$. $a \in (-1, 4)$

$B$. One root less than $3$ and other greater than $3$ $Q$. $a \in (-\infty, -1)$

$C$. One root less than $1$ and other greater than $3$ $R$. $a \in (-\infty, -4/3)$

Column-$I$	Column-$II$
$A$. Imaginary roots	$P$. $a \in (-1, 4)$
$B$. One root less than $3$ and other greater than $3$	$Q$. $a \in (-\infty, -1)$
$C$. One root less than $1$ and other greater than $3$	$R$. $a \in (-\infty, -4/3)$

Difficult

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

If $ax^2 + bx + c < 0$ for all $x \in R$ and the expressions $cx^2 + ax + b$ and $ax^2 + bx + c$ have their extreme values at the same point $x$,then for the expression $cx^2 + ax + b$:

Similar Questions

For what set of values of $K$ do both roots of the equation $4x^2 - 20Kx + (25K^2 + 15K - 66) = 0$ lie below $2$?

The number of integral values of $m$ for which the quadratic expression $(1 + 2m)x^2 - 2(1 + 3m)x + 4(1 + m)$ is always positive for all $x \in R$ is:

If $b > a$,then the equation $(x - a)(x - b) = 1$ has

The real number $k$ for which the equation $2x^2 + 3x + k = 0$ has two distinct real roots in the interval $[0, 1]$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module