# Tag Info

• 3,319

### Find the maximum of $\frac{abc}{(4a+1)(9a+b)(4b+c)(9c+1)}$,where$a,b,c>0$

Hint Try instead to minimize $$\Phi=\frac{(4a+1)(9a+b)(4b+c)(9c+1)}{abc}$$ Compute the partial derivatives All of them being equal to $0$, the solution is immediate.
• 216k
1 vote

### Why does $1+2+3+\cdots = -\frac{1}{12}$?

Most of the answers here focus either on alternative methods for computing $\zeta(-1)$, or on why the statement "$\sum_{n=1}^{\infty} n = -1/12$" doesn't make sense. While these answers are ...
• 24.1k
1 vote

• 363k

### Analyzing the proof of Kirszbraun's theorem

The sets in $(1)$ are closed balls in $\mathbb{R}^n$, so they are compact because they are closed and bounded. The reason why $K_{\gamma}$ is non-empty for large enough $\gamma$ is because for any ...
• 171

### Bijective function from $[a, b)$ to $(a, b)$

To do this requires that you know some topology of the real numbers on $(0,1)$. In particular, you need to know about the enumeration of the rational numbers in this particular open interval. Thus let ...
• 3,319

• 2,460

### Determine if the following autonomous differential equation has periodic solutions

Hint. $$\cases{ x x' = -x y\\ y y' = x y + y^3 }$$ after addition $$\frac 12(x^2+y^2)'= \frac 14 (y^4)'$$
• 26.2k
1 vote

### Compactness, open covers and intersection

Let $A,K$ be separated subsets of a metric space. That is, $A\cap\overline K=\emptyset=\overline A \cap K.$ Then $A,K$ are completely separated. That is, there are open sets $U,V$ with $A\subset U$ ...
• 53.9k