#1 Suppose that the differentiable function $\displaystyle f(x)$ satisfies $\displaystyle f(1)=2 $ and $\displaystyle f'(x) \leq 1$. How large can $\displaystyle f(3)$ possibly be?(Done)

If $\displaystyle f(3)=4$, then show that $\displaystyle f(x)=x+1$ for $\displaystyle 1 \leq x \leq 3$.

How to approach this?

#2 Assume $\displaystyle f(x)$ is continuous on $\displaystyle [a,b]$, and for any $\displaystyle x\in[a,b]$, there exists y\in[a,b] such that $\displaystyle |f(y)| \leq \frac{1}{2}|f(x)|$. Prove that there exists a point $\displaystyle c \in [a,b]$ such that $\displaystyle f(c)=0.$