query: Use the Mean Value Theorem to prove that there exists a $c\in(0,1)$ such that $f(c) = f'(c)$
positive_passages.docid: 143354
positive_passages.text: Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
negative_passages.docid: 94489
negative_passages.text: Mean value property implies harmonicity
negative_passages.docid: 103706
negative_passages.text: If $\{f(0)\}^{2} + \{f'(0)\}^{2} = 4$ then there is a $c$ with $f(c) + f''(c) = 0$
negative_passages.docid: 41037
negative_passages.text: Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$ under some conditions
negative_passages.docid: 56127
negative_passages.text: Prove a function is constant under certain conditions
negative_passages.docid: 58041
negative_passages.text: Prove by Mean Value Theorem $\frac{x}{1+x}0$
negative_passages.docid: 90199
negative_passages.text: Prove that there exists a constant $c>0$ such that the following happens
negative_passages.docid: 1371
negative_passages.text: Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
negative_passages.docid: 150554
negative_passages.text: proof that $f'$ is continuous at $a$
negative_passages.docid: 85739
negative_passages.text: Show that $f(x) = 0$ for all $x \in [a,b]$ given $|f'(x)| \leq C|f(x)| $
negative_passages.docid: 129579
negative_passages.text: If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$
negative_passages.docid: 71857
negative_passages.text: Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
negative_passages.docid: 67738
negative_passages.text: Mean Value theorem problem?(inequality)
negative_passages.docid: 13270
negative_passages.text: Prove $\exists \xi \in (a,b):f'(\xi)<1+f^2(\xi)$
negative_passages.docid: 74433
negative_passages.text: Why does the Mean Value Theorem require a closed interval for continuity and an open interval for differentiability?
negative_passages.docid: 30510
negative_passages.text: Second mean value theorem proof
negative_passages.docid: 53726
negative_passages.text: Prove that if $f$ is a real continuous function such that $|f|\le 1$ then $|\int_{|z|=1} f(z)dz| \le 4$