query: Prove that if $f(x) > 0 \; \forall \;x \in (a, b)$, then $\int_a^b{f(x) \; \mathrm{d}x} > 0$.
positive_passages.docid: 20415
positive_passages.text: Show that the integral of a positive function is positive
negative_passages.docid: 39073
negative_passages.text: Proving that: $\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$
negative_passages.docid: 48314
negative_passages.text: Show that if a function is not negative and its integral is $0$ than the function is $0$
negative_passages.docid: 33081
negative_passages.text: If $\int_{\Bbb R^2}|f(x+iy)|\,dx\,dy<\infty$ then prove that $f\equiv 0$
negative_passages.docid: 8836
negative_passages.text: If $f$ is continuous on $[a , b]$ and $\int_a^b f(x) p(x)dx = 0$ then $f = 0$
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: 158180
negative_passages.text: An inequality of $\int_0^1 |f(x)|dx$
negative_passages.docid: 37015
negative_passages.text: Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$
negative_passages.docid: 9957
negative_passages.text: Prove if $f:[a,b] \rightarrow \mathbb{R}$ is integrable, then so is $|f|$ and $\left|\int_{a}^{b}f(x)dx\right| \leq \int_{a}^{b}|f(x)|dx$
negative_passages.docid: 16109
negative_passages.text: Prove that $ \int _a^b\:f\left(x\right)\:\:dx\:=\:\int _a^b\:f\left(a+b-x\right)\:\:dx $
negative_passages.docid: 97643
negative_passages.text: Prove that:$f(f(x)) = x^2 \implies \int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}$
negative_passages.docid: 89537
negative_passages.text: $ \int_0^x f(t)dt=\int_0^{ax}f(t)dt+ \int_0^{bx}f(t)dt$ implies $f$ constant
negative_passages.docid: 28949
negative_passages.text: Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$
negative_passages.docid: 45205
negative_passages.text: Showing that $\int_0^\infty x^{-x} \mathrm{d}x \leq 2$.
negative_passages.docid: 82588
negative_passages.text: If $\int_0^x f \ dm$ is zero everywhere then $f$ is zero almost everywhere
negative_passages.docid: 136966
negative_passages.text: Riemann integral proof $\int^b_a f(x) \, dx>0$
negative_passages.docid: 131876
negative_passages.text: Show that $\int_{a}^{b}{x^{n}f(x)dx}=0$, then $f=0$