examday

/TMUA

Let $f(x)=3-2x$. A differentiable function $g$ satisfies $g'(x)=f(x)f'(x)(x-3)$.

For this question, take “$g$ is increasing at $x$” to mean $g'(x)\ge0$.

Decide which of the following statements are true.

• 1. $x\le\dfrac32$ is a sufficient condition for $g$ to be increasing at $x$.
• 2. $x\ge3$ is a necessary condition for $g$ to be increasing at $x$.
• 3. $x\le\dfrac32$ or $x\ge3$ is a necessary and sufficient condition for $g$ to be increasing at $x$.

How many of the statements 1–3 are true?