(D) Given $f(x) = x^2 + 2bx + 2c^2$. Completing the square,we get $f(x) = (x + b)^2 + 2c^2 - b^2$. The minimum value of $f(x)$ is $2c^2 - b^2$.Given $

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(D) Given $f(x) = x^2 + 2bx + 2c^2$. Completing the square,we get $f(x) = (x + b)^2 + 2c^2 - b^2$. The minimum value of $f(x)$ is $2c^2 - b^2$.Given $g(x) = -x^2 - 2cx + b^2$. Completing the square,we get $g(x) = b^2 + c^2 - (x + c)^2$. The maximum value of $g(x)$ is $b^2 + c^2$.According to the condition $\min f(x) > \max g(x)$,we have $2c^2 - b^2 > b^2 + c^2$.Simplifying the inequality,we get $c^2 > 2b^2$.Taking the square root on both sides,we get $|c| > |b|\sqrt{2}$.

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

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If $f(x) = x^2 + 2bx + 2c^2$ and $g(x) = -x^2 - 2cx + b^2$ such that $\min f(x) > \max g(x)$,then the relation between $b$ and $c$ is

IIT 2003 All IIT

A
No real value of $b$ and $c$
B
$0 < c < b\sqrt{2}$
C
$|c| < |b|\sqrt{2}$
D
$|c| > |b|\sqrt{2}$

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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 $f(x) = x^2 + 2bx + 2c^2$ and $g(x) = -x^2 - 2cx + b^2$ such that $\min f(x) > \max g(x)$,then the relation between $b$ and $c$ is

Similar Questions

Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}+2ax+(3a+10) = 0$ such that $\alpha < 1 < \beta$. Then the set of all possible values of $a$ is:

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 $cx^2 + ax + b$:

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