Find minimum/maximum value

Question

I need to find the points on the surface $y^2 = 25 + xz$ that are closest to the origin.
I don't have any idea how to approach this.

Can someone explain to me?

Do you know about lagrange multipliers? – Git Gud Mar 02 '14 at 15:51 — Git Gud, Mar 02 '14 at 15:51

J.R. · Answer 1 · 2014-03-02T15:58:12.390

2

Hint:

You want to minimize $\|(x,y,z)\|^2=x^2+y^2+z^2$ with the constraint $y^2=25+xz$. Use Lagrange multipliers!

edited Mar 02 '14 at 15:58

answered Mar 02 '14 at 15:51

J.R.

17,904

chubakueno · Answer 2 · 2014-03-02T16:17:03.693

1

Set $x^2+y^2+z^2=d^2$. Now you have $$d^2=25+x^2+xz+z^2=25+\left(x+\frac z 2\right)^2+ 3\frac{z^2}4\ge 25 $$ With equality iff $x=z=0$ (so $y = \pm5$ )

edited Mar 02 '14 at 16:17

answered Mar 02 '14 at 16:10

chubakueno

5,623

Find minimum/maximum value

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