IPMAT Rohtak 2019 (QA) - If the minimum value of f(x) = x^2 + 2bx + 2c^2 is greater than the maximum value of g(x) = -x^2 - 2cx + b^2, then for real value of x. | PYQs + Solutions

IPMAT Rohtak 2019

Algebra

Quadratic Equations

Medium

If the minimum value of $f(x) = x^2 + 2bx + 2c^2$ is greater than the maximum value of $g(x) = -x^2 - 2cx + b^2$ , then for real value of x.

|c| > \sqrt{2}|b|

\sqrt{2}|c| > b

0 < c < \sqrt{2}b

no real value of a

Correct Option: 1

For quadratic function

f(x) = x^2 + 2bx + 2c^2

\newline

- Minimum value occurs at

x = -b

(coefficient of

x

divided by -2)

\newline

- Minimum value =

f(-b) = b^2 + (-2b^2) + 2c^2 = 2c^2 - b^2

For quadratic function

g(x) = -x^2 - 2cx + b^2

\newline

- Maximum value occurs at

x = -c

(coefficient of

x

divided by -2)

\newline

- Maximum value =

g(-c) = -c^2 + 2c^2 + b^2 = c^2 + b^2

Given that minimum of

f(x)

> maximum of

g(x)

\newline

2c^2 - b^2 > c^2 + b^2

\newline

2c^2 - c^2 > b^2 + b^2

\newline

c^2 > 2b^2

\newline

|c| > \sqrt{2}|b|

Therefore,

|c| > \sqrt{2}|b|

is the correct condition.