If f(x) = x^2 + 2bx + 2c^2 and g(x) = -x^2 - 2cx + b^2,… — JEE Main 2003 Mathematics PYQ

AIEEE 2003

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

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

Show answer and solution

Correct option: D

Solution

The minimum of f(x) occurs at x = -b, giving min f(x) = 2c² - b².
The maximum of g(x) occurs at x = -c, giving max g(x) = b² + c².

The condition min f(x) > max g(x) becomes:
2c² - b² > b² + c²
c² > 2b²
|c| > |b|√2

[AIEEE 2003]

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JEE Main 2003 Mathematics question on Complex Numbers and Quadratic Equations