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Extrema of Quadratic Functions

CAT 2025 Slot 1 · Quantitative Ability · Medium · Algebra

This is a medium Quantitative Ability question from the CAT 2025 Slot 1 paper. It tests Algebra. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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

  1. A.

    2

  2. B.

    1/2

  3. C.

    -1/2

  4. D.

    -2

Answer

B

Explanation

For f(x)=x2−4cx+8cf(x) = x^2 - 4cx + 8c, minimum occurs at x=2cx = 2c: Min(f)=(2c)2−4c(2c)+8c=−4c2+8c\text{Min}(f) = (2c)^2 - 4c(2c) + 8c = -4c^2 + 8c

For g(x)=−x2+3cx−2cg(x) = -x^2 + 3cx - 2c, maximum occurs at x=3c2x = \frac{3c}{2}: Max(g)=−(3c2)2+3c(3c2)−2c=9c24−2c\text{Max}(g) = -\left(\frac{3c}{2}\right)^2 + 3c\left(\frac{3c}{2}\right) - 2c = \frac{9c^2}{4} - 2c

Given Min(f)>Max(g)\text{Min}(f) > \text{Max}(g): −4c2+8c>9c24−2c  ⟹  0>25c24−10c  ⟹  5c2−8c<0  ⟹  c(c−85)<0-4c^2 + 8c > \frac{9c^2}{4} - 2c \implies 0 > \frac{25c^2}{4} - 10c \implies 5c^2 - 8c < 0 \implies c\left(c - \frac{8}{5}\right) < 0 Thus 0<c<850 < c < \frac{8}{5}. Among the given options, c=1/2c = 1/2 falls in this range.

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