$C^{1}(\mathbb{R})$ be the collection of continuously differentiable functions on $\mathbb{R}$.

Question

I came across the following problem:

Let $C^{1}(\mathbb{R})$ be the collection of continuously differentiable functions on $\mathbb{R}$.Let $S$=$\{f \in C^{1}(\mathbb{R}):f(0)=0,f(1)=1,|f'(x)|\leqslant 3/4, \forall x\in \mathbb{R} \}.$Then which of the following option is correct?

(a)S is empty,
(b)S is non-empty and finite,
(c)S is countably infinite,
(d)S is uncountable.

I do not know how to progress. Please help.Thanks in advance for your time.

Answer 1

According to the mean value theorem there must be a point $x\in(0,1)$ where $$ f'(x) = \frac{f(1) - f(0)}{1-0} = 1 $$ That should answer your question.