FORM GR1268 Question 13

Question:
If \(f\) is a continuously differentiable real-valued function defined on the open interval \((-1,4)\) such that \(f(3)=5\) and \(f'(x)\ge-1\) for all \(x\), what is the greatest possible value of \(f(0)\)?

(A) \(3\)     (B) \(4\)     (C) \(5\)     (D) \(8\)     (E) \(11\)

Answer:
(D)

Answer Key:
Since the question is a asking for the greatest possible value at \(x=0\),
we want the function to be increasing as much as possible from \(x=3\) to \(x=0\).

Equivalently, we want the function to be decreasing as much as possible from \(x=0\) to \(x=3\).

The largest negative slope we can have is \(f'(x)=-1\).

Assuming \(f'(x)=-1\) over the interval \((0,3)\),
\[\begin{align}
f(3)&=f(0)-1(3-0)\\
5&=f(0)-3\\
8&=f(0).
\end{align}\]