Question

The shortest distance from the curve xy=4 to the origin is...

Answer 1

the point at which the slope of the line is perpendicular to a line thru the origin

Answer 2

Is this calculus?

Answer 3

sqrt(8)
The distance between some point of the curve and the origin can be calculated in such a way \[\sqrt{x ^{2}+(f(x))^2}\]
We need to find the minimum of that function. Let's look at it's derivative:
\[(\sqrt{x^2+(f(x))^2})'=\dfrac{x^2-16/x^2}{\sqrt(x^4+16)}\]
The derivative equals zero at one point x=2/ It is the minimum as the sign changes from - to +. So the point we are tryinf to find is x=2 y=2. So the distanse is sqrt(8).

Answer 4

f(x)=y=4/x

Answer 5

ya differentiaion apparenly

Answer 6

If so,
\[x^2+y^2 ---> to \space minimize\]
constraint
\[xy=4\]

\[\nabla f=\nabla g \lambda\]
\[\nabla f=<2x,2y>\]
\[\nabla g=<y,x>\]

\[2x=y\lambda\]
\[2y=x\lambda\]
\[xy=4\]

x=2 y=2

Sqrt[2^2 +2^2]=Sqrt[8]