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 \(3\).