query: How can I prove that $f′(x) = f ′ (- x)$
positive_passages.docid: 15155
positive_passages.text: Derivative of an even function is odd and vice versa
negative_passages.docid: 82597
negative_passages.text: How to find $ f(x)$ if $f(1-f(x))=x$ for all $x$ $\in \mathbb{R}$
negative_passages.docid: 68746
negative_passages.text: What is $f(f^{-1}(A))$?
negative_passages.docid: 76610
negative_passages.text: Show that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$
negative_passages.docid: 152841
negative_passages.text: I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive.
negative_passages.docid: 108633
negative_passages.text: Why $f^{-1}(f(A)) \not= A$
negative_passages.docid: 101778
negative_passages.text: Show that there is $\eta\in(0,1)$ with $f''(\eta)=f(\eta)$
negative_passages.docid: 2072
negative_passages.text: Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$
negative_passages.docid: 58943
negative_passages.text: If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$
negative_passages.docid: 3387
negative_passages.text: Derivative of $f(x)=|x|$
negative_passages.docid: 11322
negative_passages.text: Show that $f$ vanishes identically.
negative_passages.docid: 61782
negative_passages.text: How to prove that $f(x) = \frac{-2x+1}{(2x-1)^2-1}$ is one-to-one on $(0,1)$?
negative_passages.docid: 70304
negative_passages.text: Prove there are $\xi$, $\eta$ with $f'(\xi)f'(\eta)=1$
negative_passages.docid: 22303
negative_passages.text: How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?
negative_passages.docid: 131692
negative_passages.text: How to show that $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} = f^{2}_{2n}$
negative_passages.docid: 1641
negative_passages.text: Proving $f' = - \frac{\partial{F}/\partial{x}}{\partial{F}/\partial{y}}$
negative_passages.docid: 28949
negative_passages.text: Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$