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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.

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I suggest that you think about how the Mean Value Theorem or the Fundamental Theorem of Calculus can be applied. – Jonas Meyer Dec 13 '12 at 4:12
up vote 4 down vote accepted

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.

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Thanks a lot.I have got it. Since in $S$,$|f'(x)|\leqslant 3/4$ whereas $|f'(x)|=1$,$S$ must be empty.So $(a)$ will be the correct choice. – learner Dec 13 '12 at 4:26

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