Let $f$ be a differentiable function such that $f(0)=-3$ and $f'(x)\leq 2$ for all $x$. Of the following, which is not a possible value for $f(3)$ ?

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We are given that the derivative is always less than 2.

$$\frac{dy}{dx}=\frac{\Delta y}{\Delta x} \leq \frac{2}{1}$$

From $x=0$ to $x=3$, the maximum $f$ can increase by is:

$$\frac{\Delta y}{\Delta x} \leq \frac{6}{3}$$ $$\Delta y \leq 6$$ $$\text{Max} = f(0)+6$$ $$\text{Max}= -3+6 =3$$

Since the derivative can be negative, $f$ can be any number less than $-3$. However, the maximum value it could be is $3$.