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12 votes

Minimum of $\frac{x^3}{x-6}$ for $x>6$ without using derivative?

Here is an alternative approach. Assume that $y=\frac{x^{3}}{x-6}$ has a minimum value of $y=c$ where $c\in R$. Then $\frac{x^3}{x-6}-c=0$ will have a repeated root somewhere. Define this point as $x=...
Red Five's user avatar
  • 2,060
11 votes

Minimum of $\frac{x^3}{x-6}$ for $x>6$ without using derivative?

Applying the AM-GM inequality, we have: \begin{align*} \dfrac{x^3}{x-6} &= 108+\dfrac{216}{x-6}+18(x-6)+(x-6)^2 \\ &= \begin{aligned}[t]&108+ \dfrac{27}{x-6}+\dfrac{27}{x-6}+\dfrac{27}{x-...
Wang YeFei's user avatar
  • 6,783
10 votes

What is minimum of the integral function $I(x)= \int_0^\infty \frac{1}{(1+t^x)^x} \,dt$

Substituting $u = (1 + t^x)^{-1}$ transforms the integral to $$I(x) = \frac1x \int_0^1 u^{x - \frac1x - 1} (1 - u)^{\frac1x - 1} \,du = \frac1x \mathrm{B}\left(x - \frac1x, \frac1x\right) = \frac{\...
Travis Willse's user avatar
10 votes

Minimum of $\frac{x^3}{x-6}$ for $x>6$ without using derivative?

One thing you can do is recognize that at the minimum value, $x$, the function must be locally convex, so that a small perturbation $t$ to the left and right of $x$ should result in the same value as $...
Doug's user avatar
  • 2,556
8 votes

Minimum of $\frac{x^3}{x-6}$ for $x>6$ without using derivative?

Here is a direct proof that $\forall x>6\quad f(x)\ge f(9)$, which can be first guessed by a plot: $$\begin{align}\forall h>-3,&\quad\frac{(9+h)^3}{3+h}\ge\frac{9^3}3\\ &\iff(9+h)^3\ge3^...
Anne Bauval's user avatar
  • 35.8k
5 votes

What is minimum of the integral function $I(x)= \int_0^\infty \frac{1}{(1+t^x)^x} \,dt$

$$I(x)= \int_0^\infty \frac{1}{(1+t^x)^x} \,dt=\frac{\Gamma \left(1+\frac{1}{x}\right) \Gamma \left(x-\frac{1}{x}\right)}{\Gamma(x)}$$ Take logarithms and differentiate $$\frac{I'(x)}{I(x)}=-\frac{x^2 ...
Claude Leibovici's user avatar
4 votes
Accepted

Hessian matrix determinant greater than zero in a saddle point?

The second derivative test (and the Hessian determinant) only works for bivariate functions. For functions of three or more variables, one needs to use the eigenvalues. In this case, as we have both ...
Julio Puerta's user avatar
  • 5,286
3 votes
Accepted

Maximize Frobenius norm of product under Frobenius norm constraint

First of all, it is equivalent to consider the objective function $\|AX\|_F^2$, which can be rewritten as $$ \|AX\|_F^2 = \operatorname{tr}((AX)^T(AX)) = \operatorname{tr}(X^TA^TAX) = \operatorname{...
Ben Grossmann's user avatar
2 votes
Accepted

Prove that the sequence of points given by the gradient descent algorithm converges to zero.

You have shown that $u_n \ge 0$ for $n \ge 2$. We can take advantage of that observation. For $n\ge 2$, we have $$u_{n+1}-u_n=-2u_n^3 \le 0$$ That is the sequence is decreasing. Hence, by monotone ...
Siong Thye Goh's user avatar
2 votes
Accepted

Finding dual of a linear programming problem max $c^{T}X$ such that $Ax \geq b$ and $x \leq 0$

The direction of the constraints in the dual depends on the sign of the variable in the primal. If the primal is the maximization problem, $$\max c^Tx$$ $$Ax \ge b$$ $$x \le 0$$ then the ...
Siong Thye Goh's user avatar
1 vote

What dimensions should a can have?

Use the AM-GM inequality to get $$2\pi hR + 2t\pi R^2 = \pi h R + \pi h R + 2t\pi R^2 \ge 3\pi(2th^2 R^4)^{1/3} = 3 \sqrt[3]{2\pi t V^2}$$ where $t = 1/(1-(p/100))$. Equality holds when $hR = hR = 2tR^...
ronno's user avatar
  • 11.6k
1 vote
Accepted

matroid intersection and graph orientation

As mentioned before the statement of Theorem 6.2, when we use the matroid intersection theorem, it's enough to consider sets $U$ which are closed for matroid $M_1$. In the graph orientation case, $M_1$...
Misha Lavrov's user avatar
1 vote
Accepted

Condition to have the convexity of the infimum of multi variate polynomial

$\DeclareMathOperator*{\arginf}{\arg\!\inf}\DeclareMathOperator{\dom}{dom}$Let $f$ be defined as $f : \mathcal X \times \mathcal U \to \mathbb R$, where $x \in \mathcal X$ and $u \in \mathcal U$. Then,...
mhdadk's user avatar
  • 1,428
1 vote

Finding the global minimum of a convex-like optimization problem

I would be very surprised if you could get anything beyond local minima in this general case without using (intractable) global optimization techniques. The class of quasiconvex functions are not ...
gabalz's user avatar
  • 306
1 vote
Accepted

Derivation of steepest descent direction in a Riemannian space as in Amari 1998 - Natural gradient works efficiently in learning

We want to minimize $$L(w)+\epsilon \nabla L(w)^Ta$$ This is equivalent to minimizing $$\epsilon \nabla L(w)^Ta$$ since $L(w)$ is a constant. Furthermore, it is equivalent to minimizing $$ \nabla L(w)^...
Siong Thye Goh's user avatar
1 vote

Minimum of $\frac{x^3}{x-6}$ for $x>6$ without using derivative?

In the whole domain $\inf{f(x)}=-\infty$. Let's check for $x>6$ since we have an asymptote there, by calculating a few values by hand we find out that: $f(7)=343, f(8)=256,f(9)=243,f(10)=250$. ...
B.A.M's user avatar
  • 447
1 vote

Finding solution of a nonlinear equation containing both the vector and its norm

Let $s = \dfrac{w}{\|w\|_2},\; t = \|w\|_2$. Then $$cs - u + tAs = 0 \Leftrightarrow (cI+tA)s = u$$ Suppose A is diagonalizable. Let $(\lambda_i)$ be the eigenvalues of $A$ and $(v_i)$ be the ...
ioveri's user avatar
  • 1,517
1 vote

Difference between optimization and estimation

Optimization means selecting "a best element with regard to some criterion" (source: Wikipedia). Estimation requires a generative probability model (source: "Data analysis recipes: ...
Alexander Ness's user avatar

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