CAT 2022 Slot 2 · Quant · Question 20

A quadratic that is never negative sits entirely on or above the horizontal axis.

But f\left( 2 \right)=0, so it does touch the axis at x=2.

If the curve crossed there, it would dip below on one side, which the condition forbids.

So the curve only touches, which means x=2 is a repeated root.

Therefore f has the form f\left( x \right)=a{\left( x-2 \right)}^{2} for some positive constant a.

Using the second value, f\left( 4 \right)=a{\left( 4-2 \right)}^{2}=4a.

Setting 4a=6 gives a=\frac{3}{2}.

Now evaluate at the required point: f\left( -2 \right)=\frac{3}{2}{\left( -2-2 \right)}^{2}=\frac{3}{2}\times 16.

That is 24 — the third option.

Shortcut. With a double root at 2, the value of f depends only on the distance from 2. Since -2 is 4 away while 4 is only 2 away, f\left( -2 \right) is {2}^{2}=4 times f\left( 4 \right), that is 4\times 6=24 — no constant needed.