(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 2 Complex Numbers and Quadratic Equations: Quadratic Equations 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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2 Complex Numbers and Quadratic Equations: Quadratic Equations

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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

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Let the set of all values of $p \in R$,for which both the roots of the equation $x^2-(p+2)x+(2p+9)=0$ are negative real numbers,be the interval $(\alpha, \beta]$. Then $\beta-2\alpha$ is equal to

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