Minimum value of composite function expression

CAT 2021 Slot 3 · QA · Easy · Functions

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If f(x)=x2−7xf(x) = x^2 - 7x and g(x)=x+3g(x) = x + 3, then the minimum value of f(g(x))−3xf(g(x)) - 3x is

  1. A.
    • 20
  2. B.
    • 15
  3. C.
    • 12
  4. D.
    • 16
Answer

D

Explanation

First, evaluate f(g(x))f(g(x)): f(g(x))=(x+3)2−7(x+3)=x2+6x+9−7x−21=x2−x−12f(g(x)) = (x+3)^2 - 7(x+3) = x^2 + 6x + 9 - 7x - 21 = x^2 - x - 12

Now, f(g(x))−3xf(g(x)) - 3x: E(x)=(x2−x−12)−3x=x2−4x−12E(x) = (x^2 - x - 12) - 3x = x^2 - 4x - 12

Completing the square: E(x)=(x−2)2−16E(x) = (x - 2)^2 - 16

The minimum value occurs at x=2x = 2, which is −16-16.

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