The function f(x) = 6 - 6x - 2x^2

The correct answer is Option (3) → Strictly decreasing for \(x \gt -\displaystyle\frac{3}{2}\)

The sign of f′(x) decides where f increases or decreases:

    \(f'(x) = -6 - 4x = -2(2x + 3)\)

    \( \displaystyle f'(x) \lt 0 \iff 2x + 3 \gt 0 \iff x \gt -\frac{3}{2} \)

    \( \displaystyle f'(x) \gt 0 \iff x \lt -\frac{3}{2} \)

So f is strictly decreasing for \(x \gt -\frac{3}{2}\) and strictly increasing for \(x \lt -\frac{3}{2}\). Options (1), (2) and (4) do not fit this.