Question

If \(f(x)=-3x^2\), then f(x) is:

• A. Increasing in (0, ∞), decreasing in (-∞, 0)
• B. Increasing in (-∞, 0), decreasing in [0, ∞)
• C. Increasing in \([-\frac{1}{3},∞)\), decreasing in \((-∞,\frac{-1}{3}]\)
• D. Decreasing for all real values of x

Answer 1

The Correct Option is B

To determine the behavior of the function \(f(x)=-3x^2\), we consider its derivative to analyze intervals of increase and decrease. 

The derivative is:

\(f'(x)=-6x\).

1. Set the derivative equal to zero to find critical points:

\(-6x=0\) ⟹ \(x=0\).

2. Analyze the sign of \(f'(x)\) on the intervals divided by this critical point:

- For \(x<0\), \(f'(x)=-6x>0\). Thus, \(f(x)\) is increasing.

- For \(x>0\), \(f'(x)=-6x<0\). Thus, \(f(x)\) is decreasing.

3. Conclusion:

\(f(x)\) is increasing in \((-∞,0)\) and decreasing in \([0,∞)\).