query: How to show that If $f(x+y)=f(x)f(y)$ then $f(x)\geq 0$
positive_passages.docid: 145265
positive_passages.text: A non-zero function satisfying $g(x+y) = g(x)g(y)$ must be positive everywhere
negative_passages.docid: 59063
negative_passages.text: $f(f(x))=f(x)$ question
negative_passages.docid: 32506
negative_passages.text: Prove $kf(x)+f'(x)=0 $ when conditions of Rolle's theorem are satisfied .
negative_passages.docid: 8179
negative_passages.text: If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
negative_passages.docid: 72856
negative_passages.text: Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
negative_passages.docid: 26219
negative_passages.text: Suppose f is a function such that f'(x) = 1/x for all x > 0. Prove that if f(1) = 0, then f(xy) = f(x) + f(y) for all x,y > 0
negative_passages.docid: 14952
negative_passages.text: Prove: $\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq \sqrt{x}+\sqrt{y}$
negative_passages.docid: 135693
negative_passages.text: Suppose f is a twice differentiable function with $f(0)=0$ and $f(1)=1$, $f'(0)=f'(1)=0$. Then show that $|f''(x)| > 4$ for some $x$ in $(0,1)$
negative_passages.docid: 28088
negative_passages.text: When is $f_{xy}(x,y)\neq f_{yx}(x,y)?$
negative_passages.docid: 74810
negative_passages.text: A function $f$ such that $f' = 0$ when $x < 0$ and $f' = 1$ when $x \geq 0$?
negative_passages.docid: 31732
negative_passages.text: If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$
negative_passages.docid: 409
negative_passages.text: functions satisfying $f(x - y) = f(x) f(y) - f(a - x) f(a + y)$ and $f(0)=1$
negative_passages.docid: 69008
negative_passages.text: $f'(x) = [f(x)]^{2}$. Prove $f(x) = 0 $
negative_passages.docid: 57026
negative_passages.text: $f(0)=f'(0)=f'(1)=0$ and $f(1)=1$ implies $\max|f''|\geq 4$
negative_passages.docid: 55016
negative_passages.text: Prove that if ($d(x,y)=0$ iff $x=y$) and if $d(x,z)\leq d(x,y)+d(z,y)$ then $d$ is a metric
negative_passages.docid: 111756
negative_passages.text: How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
negative_passages.docid: 33247
negative_passages.text: Does $f(0)=0$ and $\left|f^\prime(x)\right|\leq\left|f(x)\right|$ imply $f(x)=0$?