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CAT 2004 — QA Question 12

Basics (Functions)Easy

Let f(x) = ax2 – b|x|, where a and b are constants. Then at x = 0, f(x) is

Answer & solution

  • A

    maximized whenever a > 0, b > 0

  • B

    maximized whenever a > 0, b < 0

  • C

    minimized whenever a > 0, b > 0

  • minimized whenever a > 0, b < 0

Solution

f(x) = ax2 – b|x|

x2 and |x| both are positive. Let x ≠ 0.

At x = 0, f(0) = 0

Consider the following cases:

1. a > 0, b > 0
∴ f(x) > 0, when ax2 > b|x|
∴ f(x) < 0, when  ax2 < b|x|
f(x) is neither maximised or minimised when x = 0.

2. a > 0, b < 0
∴ f(x) = ax2 + |b||x| > 0
Thus f(x) is greater than 0 when x ≠ 0.
​​​​​​​∴ f(x) is minimised at x = 0 whenever a > 0, b < 0.

Hence, option (d).

CAT 2004 QA Q12: Let f ( x ) = ax 2 &ndash; b | x |, where a and b are constants. Then at x = 0, f ( x ) is — Solution | TheCATExam