query: If $f \in \mathcal C^1[a,b]$, $f(a)=0$ and supose there is a $\lambda >0$ such that $|f'(x)| \leq \lambda |f(x)| \forall x \in (a,b)$
positive_passages.docid: 145573
positive_passages.text: How to show that $f$ is a zero function?
negative_passages.docid: 158561
negative_passages.text: Let $f (z) $ be an entire function such that $|f (z)|≤K|z|$, $∀z∈\mathbb{C}$, for some $K>0$. If $f (1) =i$, then$f (i) $ is
negative_passages.docid: 108740
negative_passages.text: $C^1[a,b]$ is closed in $C[a,b]$
negative_passages.docid: 7825
negative_passages.text: If $f$ is a concave function and $f(0) \geq 0$ then $f(a+b) \leq f(a) + f(b)$ for all $a,b >0$
negative_passages.docid: 90199
negative_passages.text: Prove that there exists a constant $c>0$ such that the following happens
negative_passages.docid: 101286
negative_passages.text: How show that $|a_{n}-1|\le c\lambda ^n,\lambda\in (0,1)$
negative_passages.docid: 12556
negative_passages.text: A non-trivial, non-negative, function bounded below by its derivative with $f(0)=0$?
negative_passages.docid: 99458
negative_passages.text: Suppose $\int_{[a,b]}f=0,\text{ then }f(x)=0 \forall x\in[a,b]$
negative_passages.docid: 84313
negative_passages.text: $|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$
negative_passages.docid: 13575
negative_passages.text: Need help for proving that: $f(f^{-1}(A)) ⊆ A$
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: 101778
negative_passages.text: Show that there is $\eta\in(0,1)$ with $f''(\eta)=f(\eta)$
negative_passages.docid: 136980
negative_passages.text: Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.
negative_passages.docid: 94781
negative_passages.text: Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
negative_passages.docid: 47026
negative_passages.text: if $f^{(n)}(x_0),g^{(n)}(x_0)$ exist and $\lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n}=0$ then $f^{(r)}(x_0)=g^{(r)}(x_0 )$ for all $0\leq r\leq n.$
negative_passages.docid: 132013
negative_passages.text: How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
negative_passages.docid: 30694
negative_passages.text: Show that there exist $[a,b]\subset [0,1]$, such that $\int_{a}^{b}f(x)dx$ = $\int_{a}^{b}g(x)dx$ = $\frac{1}{2}$