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.

For quadratic function $f(x) = x^2 + 2bx + 2c^2$:

• Minimum value occurs at $x = -b$ (coefficient of $x$ divided by -2)
  • Minimum value = $f(-b) = b^2 + (-2b^2) + 2c^2 = 2c^2 - b^2$

For quadratic function $g(x) = -x^2 - 2cx + b^2$:

• Maximum value occurs at $x = -c$ (coefficient of $x$ divided by -2)
  • Maximum value = $g(-c) = -c^2 + 2c^2 + b^2 = c^2 + b^2$

Given that minimum of $f(x)$ > maximum of $g(x)$:

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

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

$c^2 > 2b^2$

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

Therefore, $|c| > \sqrt{2}|b|$ is the correct condition.