Tagged Questions
7
votes
1answer
138 views
Prove that $f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$
Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that
$$f(x^2)+f(y^2)\le2 f(\sqrt{x y}), \space x,y\ge0 $$
Prove that
$$f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$$
Where should ...
14
votes
1answer
155 views
How to prove $\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $?
I want to prove the inequality
$$\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $$
There are some obstacles I face: the indefinite integral cannot be expressed in terms of ...
2
votes
4answers
100 views
prove or disprove that $\int_a^b |f(x)| \mathrm{d}x\geq |\int_a^b f(x)\mathrm{d}x |$
+Let $f$ be a continuous and integrable function over $[a;b]$, Prove or disprove that :
$\displaystyle\int_a^b |f(x)|\ \mathrm{d}x\geq \left | \int_a^b f(x)\ \mathrm{d}x\right|
$
1
vote
1answer
109 views
Show that $\int_0^1f(x)^3dx\le\left(\int_0^1f(x)dx\right)^2$ [duplicate]
Possible Duplicate:
Prove that $\int_0^x f^3 \le \left(\int_0^x f\right)^2$
Let $f$ be a differentiable function on $[0,1]$. $f(0)=0$ and $1\ge f'(x)\ge0$. Show that
...
9
votes
1answer
314 views
Prove that $\int_0^1|f''(x)|dx\ge4.$
Let $f$ be a $C^2$ function on $[0,1]$. $f(0)=f(1)=f'(0)=0,f'(1)=1.$ Prove that
$\int_0^1|f''(x)|dx\ge4.$
Also determine all possible $f$ when equality occurs.
6
votes
1answer
146 views
Integral-Summation inequality.
The following question was in an entrance exam:
Show that, if $n\gt0$, then: $$\int_{{\rm e}^{1/n}}^{\infty}{\frac{\ln{x}}{x^{n+1}}\:dx}=\frac{2}{n^2{\rm e}}$$ You are allowed to assume ...
6
votes
2answers
298 views
Prove that $ 1.462 \le \int_0^1 e^{{x}^{2}}\le 1.463$
Prove the following integral inequality:
$$ 1.462 \le \int_0^1 e^{{x}^{2}}\le 1.463$$
This is a high school problem. So far i did manage to prove that the integral is bigger than $1.462$ by using ...
6
votes
4answers
576 views
Jensen's inequality for integrals
What nice ways do you know in order to prove Jensen's inequality for integrals? I'm looking for some various approaching ways.
Supposing that $\varphi$ is a convex function on the real line and $g$ is ...
7
votes
1answer
374 views
Prove that: $ \int_{0}^{1} \ln \sqrt{\frac{1+\cos x}{1-\sin x}}\le \ln 2$
I plan to prove the following integral inequality:
$$ \int_{0}^{1} \ln \sqrt{\frac{1+\cos x}{1-\sin x}}\le \ln 2$$
Since we have to deal with a convex function on this interval i thought of ...
3
votes
3answers
132 views
Squeeze an integral
Would you have any idea about this problem ?
Prove that for all nonnegative integers $n$, the following inequalities hold:
$$\frac{e^2}{n+3}\leq \int_{1}^{e} x (\ln x)^n \,dx \leq ...
