Question

Let \( f(x) = ax^2 - b |x| \), where \(a\) and \(b\) are constants. Then at \(x = 0\), \(f(x)\) is

• A. maximized whenever \( a>0, b>0 \)
• B. maximized whenever \( a>0, b<0 \)
• C. minimized whenever \( a>0, b>0 \)
• D. minimized whenever \( a>0, b<0 \)

Hint:

When dealing with absolute values in quadratic functions, consider the behavior of the absolute term at the critical point.

Answer 1

The Correct Option is C

The function \( f(x) = ax^2 - b |x| \) involves an absolute value function. - When \(x = 0\), the function simplifies to \(f(0) = 0\). - If \(a>0\) and \(b>0\), the quadratic term dominates, leading to a minimum at \(x = 0\). Thus, the function is minimized when \(a>0\) and \(b>0\). \[ \boxed{\text{Minimized whenever } a>0, b>0} \]