Let f be a differentiable function such that f(0) = -5 and f'(x) is less than or equal to 3 for all x. Of the following, which is not a possible value for f(2)?
A) -10 B) -5 C) 0 D) 1 E) 2

Question

Let f be a differentiable function such that f(0) = -5 and f'(x) is less than or equal to 3 for all x. Of the following, which is not a possible value for f(2)?
A) -10   B) -5   C) 0   D) 1   E) 2

anonymous · Answer

Let's suppose $f'(x)=3$ for all $x$ (the maximum given). By the mean value theorem, over some interval $(0,k)$, with $k>0$, there is some $c$ such that
$$f'(c)=3=\frac{f(k)-f(0)}{k-0}=\frac{f(k)+5}{k}~~\implies~~3k-5=f(k)$$
Now if $k=2$, we would get $f(k)=1$. What does this suggest?

anonymous · Answer

I'm not sure but I think it means that f(2) is less than or equal to 1?

anonymous · Answer

Right! We choose the maximum value for $f'(x)$ to guarantee an upper bound for the value of $f(k)$, in this case $f(2)$. We know that $f(2)\le1$, which means $f(2)$ cannot possibly be $2$.

anonymous · Answer

Oh, okay. Thanks!

anonymous · Answer

You're welcome!

error1603 · Answer

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