Intereting Posts

homomorphism $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ with multiplicative groups, prove that kernel of $f$ is infinite.
Do results from any $L^p$ space for functions hold in the equivalent $\ell^p$ spaces for infinite sequences?
Finite number of subgroups $\Rightarrow$ finite group
For which values of $\alpha$ and $\beta$ does the integral $\int\limits_2^{\infty}\frac{dx}{x^{\alpha}ln^{\beta}x}$ converge?
Verify this identity: $\sin x/(1 – \cos x) = \csc x + \cot x$
Are there any Symmetric Groups that are cyclic?
Isomorphisms (and non-isomorphisms) of holomorphic degree $1$ line bundles on $\mathbb{CP}^1$ and elliptic curves
Can $\sin(\pi/25)$ be expressed in radicals
Evaluate $ \int_0^\pi \left( \frac{2 + 2\cos (x) – \cos((k-1)x) – 2\cos (kx) – \cos((k+1)x)}{1-\cos (2x)}\right) \mathrm{d}x $
A formula $\phi$ is logically equivalent to a another formula which contains only propositional variables and the connectives $\wedge$ and $\to$
A and B are nxn matrices, $A =B^TB$. Prove that if rank(B)=n, A is pos def, and if rank(B)<b, A is pos semi-def.
Representations of integers by a binary quadratic form
A polynomial that is zero on an open set
Infinitely many systems of $23$ consecutive integers
When does equality hold in the Minkowski's inequality $\|f+g\|_p\leq\|f\|_p+\|g\|_p$?

Finding the dimensions of the maximum volume box inside the ellipsoid.

I assume that the volume of a box, $V(x,y,z) = xyz$ (they did not give this to me, but this is the volume of a box right?)

Ellipsoid:

- Shortest distance between two general curves using matlab
- Intuitive understanding of the derivatives of $\sin x$ and $\cos x$
- Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$
- Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\,\epsilon>0$ converge?
- please solve a 2013 th derivative question?
- Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$

and I use Lagrange multipliers to find an incorrect answer, I end up getting

$$x = \frac{\sqrt{a}}{\sqrt{3}}$$

$$y = \frac{\sqrt{b}}{\sqrt{3}}$$

$$z = \frac{\sqrt{c}}{\sqrt{3}}$$

the hint they give me is that

$$\text{Max volume} = \frac{8abc}{3\sqrt{3}}$$

Could someone tell me where I am doing this wrong?

- Summation of n-squared, cubed, etc.
- Why is the composition of smooth multivariable functions smooth?
- Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$
- How to calculate the asymptotic expansion of $\sum \sqrt{k}$?
- Convergence of double series $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{\sin(\sin(nm))}{n^2+m^2}$
- An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$.
- The Intuition behind l'Hopitals Rule
- average value theorem and calculus integration
- Proof that derivative of a function at a point is the slope of the tangent at the point
- Evaluate the following integral $\int_{0}^{10}\sqrt{-175e^{-t/4}+400}dt$

Probably the ellipsoid is

$$

\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1

$$

and your solution becomes $x=a/\sqrt{3}$, $y=b/\sqrt{3}$, $z=c/\sqrt{3}$ which gives the correct volume (remember to multiply by $8$, because $x$, $y$ and $z$ are half the sides of the box).

Here is a solution without calculus.

As shown in an answer to a similar question some inequalities between means can be useful here. For example, inequality between geometric mean and arithmetic mean or inequality between geometric mean and quadratic mean (a.k.a. root mean square).

We know that for any real numbers $x_{1,2,3}\ge0$ we have

$$x_1x_2x_3 \le \left(\frac{x_1+x_2+x_3}3\right)^3. \tag{1}$$

For $x_1=x^2/a^2$, $x_2=y^2/b^2$ and $x_3=z^3/c^3$ we get

$$\frac{x^2y^2z^2}{a^2b^2c^2} \le \frac1{27}.$$

Since the volume is $V=8xyz$, we have

$$V^2 \le \frac{8^2a^2b^2c^2}{27},$$

i.e.

$$V \le \frac{8abc}{3\sqrt3}.$$

Moreover, equality in (1) is attained only for $x_1=x_2=x_3=\frac{x_1+x_2+x_3}3$ which, in our case, gives $\frac xa = \frac yb = \frac zc = \frac1{\sqrt3}$. (We get this from $x_1=x_2=x_3=\frac{x^2}{a^2}=\frac{y^2}{b^2}=\frac{z^2}{c^2}=\frac13$.)

For more about inequalities of various mean see, for example:

- Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality at AoPS
- Inequality of arithmetic and geometric means at Wikipedia
- Proofs of AM-GM inequality

- Stirling number of the first kind: Proof of Recursion formula
- Are the $\mathcal{C}^k$ functions dense in either $\mathcal{L}^2$ or $\mathcal{L}^1$?
- Finding Sylow 2-subgroups of the dihedral group $D_n$
- Is polynomial $1+x+x^2+\cdots+x^{p-1}$ irreducible?
- closed form of $\int_{0}^{2\pi}\frac{dx}{(a^2\cos^2x+b^2\sin^2x)^n}$
- Show that a positive operator is also hermitian
- Intuition explanation of taylor expansion?
- Prove that in an ordered field $(1+x)^n \ge 1 + nx + \frac{n(n-1)}{2}x^2$ for $x \ge 0$
- The space obtained by identifying the antipodal points of a circle
- Books like Grundlagen der Analysis in French
- Closed model categories in the sense of Quillen vs the modern sense
- Is a polynomial equation of degree $\ge 5$ not solvable by any way?
- how to derive the mean and variance of a Gaussian Random variable?
- $66$ points in $100$ shots.
- How is $A\sin\theta +B\cos\theta = C\sin(\theta + \phi)$ derived?