query: Prove that $e^x \gt 0$ for $x \in \mathbb{R}$
positive_passages.docid: 21923
positive_passages.text: Prove that $\exp(x)>0$ using only formal definition of exp
negative_passages.docid: 8360
negative_passages.text: $\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$
negative_passages.docid: 98282
negative_passages.text: Prove that the function $\xi\in R \mapsto {e^{i\cdot \xi\cdot λ}-1\over i\cdot \xi}-λ$ is $C^{\infty}$
negative_passages.docid: 139288
negative_passages.text: Formal proof that $e^x$ is not algebraic
negative_passages.docid: 132957
negative_passages.text: Prove that the derivative of $x^w$ is $w x^{w-1}$ for real $w$
negative_passages.docid: 75653
negative_passages.text: Prove that $|f ''|\ge 4$
negative_passages.docid: 136980
negative_passages.text: Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.
negative_passages.docid: 76148
negative_passages.text: Show that $\exists c \in \mathbb{R} : f(c) = 0$
negative_passages.docid: 118091
negative_passages.text: Taylor expansion of $\exp(-x)$: proof tail is positive
negative_passages.docid: 67851
negative_passages.text: Does $|x|^p$ with $0<p<1$ satisfy the triangle inequality on $\mathbb{R}$?
negative_passages.docid: 158681
negative_passages.text: How to show that $e^x$ is differentiable?
negative_passages.docid: 136195
negative_passages.text: Stuck trying to prove that $e^{-x^{-2}}$ is $C^{\infty}$
negative_passages.docid: 142922
negative_passages.text: How to prove that $e^x$ is convex?
negative_passages.docid: 116387
negative_passages.text: How to prove that $\Gamma(x) \leq x^x$ for $x \geq 1$
negative_passages.docid: 123806
negative_passages.text: How do I show that the integral of $e^{inx}$ over a set of measure $1$ is nonzero for some nonzero $n$?
negative_passages.docid: 150502
negative_passages.text: If $f(0) = 0$ and $|f'(x)|\leq |f(x)|$ for all $x\in\mathbb{R}$ then $f\equiv 0$
negative_passages.docid: 88328
negative_passages.text: Proof of $1+x\leq e^x$ for all x?