query: Is $f ≡ 0$ if $f(0)=0$ and $f'(x) \le f(x)$?
positive_passages.docid: 28949
positive_passages.text: Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$
negative_passages.docid: 111756
negative_passages.text: How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
negative_passages.docid: 145573
negative_passages.text: How to show that $f$ is a zero function?
negative_passages.docid: 95288
negative_passages.text: Prove that, $f'(0) \ge -\sqrt{2}$ for a function $f$ satisfying some conditions on $(-1,1)$.
negative_passages.docid: 21954
negative_passages.text: A question about $f(x)\equiv 0$
negative_passages.docid: 33247
negative_passages.text: Does $f(0)=0$ and $\left|f^\prime(x)\right|\leq\left|f(x)\right|$ imply $f(x)=0$?
negative_passages.docid: 113343
negative_passages.text: $f$ is constant if derivative equals zero
negative_passages.docid: 25467
negative_passages.text: Prove that $\lim f(x) =0$ and $\lim (f(2x)-f(x))/x =0$ imply $\lim f(x)/x =0$
negative_passages.docid: 33989
negative_passages.text: If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$.
negative_passages.docid: 148737
negative_passages.text: Is $|xy|$ differentiable in $(0,0)$?
negative_passages.docid: 69008
negative_passages.text: $f'(x) = [f(x)]^{2}$. Prove $f(x) = 0 $
negative_passages.docid: 23465
negative_passages.text: When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?
negative_passages.docid: 147325
negative_passages.text: Proving that if $f(x)=x$ has no real solution then $f(f(x))=x$ has no real solution either
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: 106830
negative_passages.text: are $f(f(x))=f(x)$ and $f(x)=x$ equivalent?
negative_passages.docid: 74583
negative_passages.text: Prove that $f'(0)=L$.
negative_passages.docid: 107727
negative_passages.text: $m\{x\in [0,1]:f'(x)=0\}>0$