query: If $f'$ is increasing and $f(0)=0$, then $f(x)/x$ is increasing
positive_passages.docid: 150699
positive_passages.text: Show that if $f'$ is strictly increasing, then $\frac{f(x)}{x}$ is increasing over $(0,\infty)$
negative_passages.docid: 17154
negative_passages.text: Does a non-trivial solution exist for $f'(x)=f(f(x))$?
negative_passages.docid: 64339
negative_passages.text: Find $f$ where $f'(x) = f(1+x)$
negative_passages.docid: 74583
negative_passages.text: Prove that $f'(0)=L$.
negative_passages.docid: 130322
negative_passages.text: Show that $f(x) = 0$ if $x = 0$ $f(x) = \text{exp}(-1/x^{2})$ if $x \neq 0$ is infinitely differentiable but not analytic at $x_{0} = 0$ `
negative_passages.docid: 44060
negative_passages.text: If $\lim_{x \to +\infty} f'(x) = L$, then $\lim_{x \to \infty} \frac {f(x)}{x} = L$
negative_passages.docid: 8854
negative_passages.text: Prove $f $ is identically zero if $f(0)=0$ and $|f'(x)|\le|f(x)|$
negative_passages.docid: 91852
negative_passages.text: A continuous, injective function $f: \mathbb{R} \to \mathbb{R}$ is either strictly increasing or strictly decreasing.
negative_passages.docid: 26207
negative_passages.text: If $f$ is a quadratic and $f(x)>0\;\forall x$, and $g= f + f' + f''$, prove $g(x)>0\; \forall x$
negative_passages.docid: 105658
negative_passages.text: Show that a function that is locally increasing is increasing?
negative_passages.docid: 113343
negative_passages.text: $f$ is constant if derivative equals zero
negative_passages.docid: 17646
negative_passages.text: If $\frac{f(x)}{x}$ is decreasing in $(0,\infty)$, prove that $f(a+b) \leq f(a) + f(b)$ for $a,b >0$
negative_passages.docid: 121771
negative_passages.text: Proof $f(x)\equiv 0$
negative_passages.docid: 79301
negative_passages.text: Could a constant function be considered increasing?
negative_passages.docid: 74673
negative_passages.text: True or false: if $f'(c)<0$, then $f$ is concave down at $x=c$?
negative_passages.docid: 21879
negative_passages.text: Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals A
negative_passages.docid: 2717
negative_passages.text: Help proving existence of an specific $x_0$ for which $f''(x_0)=0$