Intereting Posts

Let $p$ be an odd prime number. How many $p$-element subsets of $\{1,2,3,4, \ldots, 2p\}$ have sums divisible by $p$?
A real vector space is an inner product space if every two dimensional subspace is an inner product space ?
Practical system with the following ODE form
Recursive Function – $f(n)=f(an)+f(bn)+n$
Motivation for the mapping cone complexes
How can I show that a sequence of regular polygons with $n$ sides becomes more and more like a circle as $n \to \infty$?
Proof that if an algebraic integer is rational, it is integer?
Fundamental unit in the ring of integers $\mathbb Z$
How to explain the commutativity of multiplication to middle school students?
Why do maximal-rank transformations of an infinite set $X$ generate the whole full transformation semigroup?
Integration of $x^2 \sin(x)$ by parts
Why can't the Polynomial Ring be a Field?
Example of a function such that $\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$ but $\varphi$ is not convex
Why is $|Y^{\emptyset}|=1$ but $|\emptyset^Y|=0$ where $Y\neq \emptyset$
Find the Jacobian

Let $ f $ be a continuous function defined on $ [0,\pi] $. Suppose that

$$ \int_{0}^{\pi}f(x)\sin {x} dx=0, \int_{0}^{\pi}f(x)\cos {x} dx=0 $$

Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $

- Proof that this limit equals $e^a$
- Multivariate Taylor Series Derivation (2D)
- Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?
- How to evalutate this exponential integral
- Method of Exhaustion applied to Parabolic Segment in Apostol's Calculus
- What are some easy to understand applications of Banach Contraction Principle?

- Computing limits which involve square roots, such as $\sqrt{n^2+n}-n$
- Integral of $\sqrt{1-x^2}$ using integration by parts
- Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$
- A curious equation containing an integral $\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}$
- Eventually periodic point and homeomorphism.
- Explanation of the binomial theorem and the associated Big O notation
- If a Sequence of Polynomials Converge to Another Polynomial Then the Roots Also Converge.
- Integral equation
- Is there any known relationship between $\sum_{i=1}^{n} f(i)$ and $\sum_{i=1}^{n} \dfrac {1}{f(i)}$
- Is there an integral for $\pi^4-\frac{2143}{22}$?

Here is one root:

Let $F(x) = \int_{0}^x f(t) \sin t dt$. Then $F(0)=0$ and $F(\pi)=\int_{0}^\pi f(t)\sin tdt=0$. So by the intermediate value theorem, there exists $0<c<\pi$ such that

$$

0=F'(c) = f(c)\sin c.

$$

But since $\sin c\neq 0$, we get that $f(c)=0$.

If f has only one real root on $ (0,\pi)$, say $ a \in (0,\pi) $, then define

$ g(x) = f(x) \sin(x-a) = f(x) (\sin(x)\cos(a) – \cos(x)\sin(a))$, then $ g(x) $ is either non-positive or non-negative, not identically zero, and has integral $ 0 $. Contradiction.

- Can someone tell me what group this is?
- If $A^m=I$ then A is Diagonalizable
- inverse Laplace transfor by using maple or matlab
- Trace of a real, symmetric positive semi-definite matrix
- A question about a Sobolev space trace inequality (don't understand why it is true)
- Beautiful proof for $e^{i \pi} = -1$
- How to show that the set of points of continuity is a $G_{\delta}$
- inverse of diagonal plus sum of rank one matrices
- Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups?
- What is the combinatorial proof for the formula of S(n,k) – Stirling numbers of the second kind?
- On critical points of a large function
- If every $x\in X$ is uniquely $x=y+z$ then $\|z\|+\|y\|\leq C\|x\|$
- composition of continuous functions
- Is tautological bundle $\mathcal{O}(1)$ or $\mathcal{O}(-1)$?
- Proving that for each two parabolas, there exists a transformation taking one to the other