(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

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(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

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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 $= \frac{3b}{4}$

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4-2.Quadratic Equations and Inequations

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Quadratic expressions and Position of roots

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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

Let $f(x) = x^2 + 2bx + 2c^2$ and $g(x) = -x^2 - 2cx + b^2$,where $x \in R$. If $b$ and $c$ are nonzero real numbers such that $\min f(x) > \max g(x)$,then $\left|\frac{c}{b}\right|$ lies in the interval:

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