CAT 2022 Slot 2 QA Question 15: Let $f(x)$ be a quadratic polynomial in $x$ such that $f(x) \geq 0$ for all real numbers $x$. If $f(2)=0$ and $f(4)=6$, then $f(-2)$ is equal to

Question

Let $f(x)$ be a quadratic polynomial in $x$ such that $f(x) \geq 0$ for all real numbers $x$. If $f(2)=0$ and $f(4)=6$, then $f(-2)$ is equal to

Accepted Answer

Since $f(x)$ is a quadratic polynomial and $f(x) \geq 0$ for all real $x$, any real root must be a repeated root.

As $f(2) = 0$, $x = 2$ is a repeated root. Hence:

$f(x) = a(x - 2)^2$

Using $f(4) = 6$:

$a(4 - 2)^2 = 6$
$4a = 6$
$a = \dfrac{3}{2}$

Now:

$f(-2) = \dfrac{3}{2}(-2 - 2)^2$
$= \dfrac

Let f(x)f(x)f(x) be a quadratic polynomial in xxx such that f(x)≥0f(x) \geq 0f(x)≥0 for all real numbers xxx. If f(2)=0f(2)=0f(2)=0 and f(4)=6f(4)=6f(4)=6, then f(−2)f(-2)f(−2) is equal to