Find Minimum Value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$

In summary, the purpose of finding the minimum value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$ is to determine the lowest possible value that the expression can have. To find the minimum value, techniques such as differentiation or completing the square can be used. The minimum value cannot be negative and there are restrictions on the values of x and y. Additionally, it is possible for the minimum value to be achieved at multiple points.
  • #1
anemone
Gold Member
MHB
POTW Director
3,883
115
Find the minimum value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$ for $0<x<\sqrt{2}$ and $y>0$.
 
Mathematics news on Phys.org
  • #2
anemone said:
Find the minimum value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$ for $0<x<\sqrt{2}$ and $y>0$.

Solution suggested by other:
The given function is the square of the distance between a point of the quarter of the circle $x^2+y^2=2$ in the open first quadrant and a point of the half hyperbola $xy=9$ in that quadrant. Then tangents to the curves at $(1,\,1)$ and $(3,\,3)$ separate the curves, and both are perpendicular to $x=y$, so those points are at the minimum distance, and the answer is $(3-1)^2+(1-3)^2=8$.
 

FAQ: Find Minimum Value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$

What is the purpose of finding the minimum value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$?

The purpose is to determine the lowest possible value that the expression can have, which can be useful in optimization problems or in finding the equilibrium point of a system.

How do you find the minimum value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$?

To find the minimum value, we can use techniques such as differentiation or completing the square and setting the derivative or completed expression equal to 0.

Can the minimum value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$ be negative?

No, the minimum value cannot be negative because it is the square of a real number, and squares are always non-negative.

Are there any restrictions on the values of x and y in $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$?

Yes, there are restrictions on the values of x and y. For the expression to be defined, x must be between -√2 and √2, and y must be non-zero.

Can the minimum value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$ be achieved at multiple points?

Yes, it is possible for the minimum value to be achieved at multiple points, especially if the expression contains absolute values or has multiple local minimums.

Similar threads

MHB Prove Inequality x,y with $x+y=2$
Replies
1
Views
895
MHB Inequality in Triangle: Prove $|x^2+y^2+z^2-2(xy+yz+xz)|\le \frac{1}{27}$
Replies
1
Views
1K
MHB Maximum and minimum of a function
Replies
1
Views
869
MHB Graph of $y=\sin{x}-2$ on the domain $[0,2\pi]$
Replies
2
Views
981
MHB Finding Min Value of $\dfrac{|b|+|c|}{a}$ from Roots of Cubic Equations
Replies
4
Views
1K
MHB Solving System of Equalities: x^2-y\sqrt xy & y^2-x\sqrt xy =3
Replies
2
Views
870
MHB Min Value of $\dfrac{a+3c}{a+2b+c}$+$\dfrac{4b}{a+b+2c}$+$\dfrac{8c}{a+b+3c}$
Replies
1
Views
1K
MHB Finding minimum value of function with two variable
Replies
7
Views
2K
MHB What is the Trigonometric Inequality for $0<x<\dfrac{\pi}{2}$?
Replies
1
Views
885
MHB Min Value of $a^2+b^2$ in Quadratic Equation
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top