1
vote
1answer
57 views

Absolute extrema of a multivariable function bounded by an ellipse

I have a function $f(x,y) = 2x + x^2 + y^2$ bounded by the ellipse $x^2 + 4y^2 \leq 24$ I know how to determine the extrema within the ellipse by getting the partial derivatives and setting them to ...
14
votes
4answers
537 views

How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$

let $a,b,c,d,e\in R$,and such $$a^2+b^2+c^2+d^2+e^2=1$$ find this value $$A=\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$$ I use computer have this $$A=\dfrac{2}{\sqrt{10}}$$ ...
1
vote
1answer
70 views
2
votes
2answers
114 views

How to prove this max absolute value equation?

How to prove this equation? $$\max(|x_1-x_2|,|y_1-y_2|) = \frac{\left|x_1+y_1-x_2-y_2\right|+\left|x_1-y_1-(x_2-y_2)\right|}{2}$$
1
vote
2answers
581 views

How to find minimum of sum of mod functions?

How to find minimum value of $$|x-1| + |x-2| + |x-31| + |x-24| + |x-5| + |x-6| + |x-17| + |x-8| + \\|x-9| + |x-10| + |x-11| + |x-12|$$ and also where it occurs ? I know the procedure for find answer ...
1
vote
1answer
54 views

What is the minimum of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ over all complex numbers?

Find the Least value of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ My try:: Let $A(2,2)$ and $B(-1,5)$ and $C(6,-2)$ and $P(x,y)$ be a point Here $A,B$ and $C$ are the point ...
1
vote
2answers
96 views

Finding absolute max and min values of function

Function given as $f(x,y) = 3x^2 + 2xy^2$. If $(x,y)$ lies in the region inside including edges of the triangle in the first quadrant given by $x\ge0, y\ge0, y\le2-x$. Reduce $f$ to a single variable ...
1
vote
1answer
46 views

minimum of absolute value

If we consider the following problem $$ \mathbb{E}[(Y-y)^2 | X=x] $$ I can easily show that the minimum with respect to $y$ occurs at $$ y=\mathbb{E}[Y |X=x] $$ How can I find the minimum of $$ ...
2
votes
2answers
519 views

How to minimize an equation with absolute values?

How would I go about minimizing the expression $\left(|z_1| + |z_2|\right) \times \left(|z_1 + z_2| + |z_1 - z_2|\right)$ subject to the constraint $|z_1|^2 + |z_2|^2 = 1$ given that $z_1$ and ...