CAT 2025 Slot 1 QA Question 20: A value of $c$ for which the minimum value of $f(x) = x^2 - 4cx + 8c$ is greater than the maximum value of $g(x) = -x^2 + 3cx - 2c$, is

Question

A value of $c$ for which the minimum value of $f(x) = x^2 - 4cx + 8c$ is greater than the maximum value of $g(x) = -x^2 + 3cx - 2c$, is

Accepted Answer

The minimum or maximum value of a quadratic occurs at its vertex.
For $f(x) = x^{2} - 4cx + 8c$, the coefficient of $x^{2}$ is positive, so it has a minimum.
$x_f = \dfrac{-(-4c)}{2} = 2c$
$f_{\min} = (2c)^2 - 4c(2c) + 8c$
$f_{\min} = -4c^2 + 8c$

For $g(x) = -x^{2} + 3cx - 2c$, the coefficient of $

A value of ccc for which the minimum value of f(x)=x2−4cx+8cf(x) = x^2 - 4cx + 8cf(x)=x2−4cx+8c is greater than the maximum value of g(x)=−x2+3cx−2cg(x) = -x^2 + 3cx - 2cg(x)=−x2+3cx−2c, is

A value of $c$ for which the minimum value of $f(x) = x^2 - 4cx + 8c$ is greater than the maximum value of $g(x) = -x^2 + 3cx - 2c$ , is