Functional on $C^1[a,b]$

Question

Let $C^1[a,b]$ be the collection of all differentiable real valued functions on $[a,b]$ with continuous derivatives and let $f$ be a functional on $C^1[a,b]$ defined by $f(x)=x'(c)$ for all $x(t)\in C^1[a,b]$ where $c=\frac{a+b}{2}$. I have to find whether $f$ is bounded or unbounded with respect to the 'sup' norm ($\|x\|=\sup\{x(t):t\in [a,b]\}$).

Answer 1

Hint. Consider the $C^1$ function $x_n(t)=\arctan(n(t-c))$ with $n\in\mathbb{N}$. What is $f(x_n)=x_n'(c)$? What may we conclude?