Stuck on what is probably a very simple problem...

College Life
1

<p>For anyone who is learning/has learned maxima and minima in Multivariable Calculus, I need help...</p>

<p>So we have the eqn, f(x,y)=√(x^2+y^2) (square root of x squared plus y squared) and we have to find the maximum/minimum point if there is such.</p>

<p>So I did all of my work here: [URL=<a href="http://imageshack.us/photo/my-images/33/maxminproblem.png/%5D%5BIMG%5Dhttp://img33.imageshack.us/img33/9456/maxminproblem.png%5B/IMG%5D%5B/URL"&gt;http://imageshack.us/photo/my-images/33/maxminproblem.png/]

http://img33.imageshack.us/img33/9456/maxminproblem.png

[/URL</a>]</p>

<p>but I need help determining how to find the second order partial derivative. I tried the quotient derivative rule but it's not working out.</p>

2

<p>I’m rusty w/ this because I took it last fall, but try writing the quotient as a product of x or y with the square root to a negative power.</p>

3

<p>fxy = -y/(x^2 + y^2)^3/2</p>

<p>fxx = 1/(x^2 + y^2)^(1/2) - x ^2/(x^2 + y^2)^3/2</p>

<p>(I haven’t got anywhere to write this down so there may be mistakes.)</p>

4

<p>Alternatively without second partial derivatives:</p>

<p>f(x,y) is obviously non-negative for all values of x and y. f(0,0) = 0. Hence f assumes its global minimum value at (0,0).</p>

5

<p>f is the distance from (0,0) to (x,y) in the R^2 plane, which clearly has a minimum value of 0 at (0,0).</p>

6

<p>[Wolfram|Alpha:</a> Computational Knowledge Engine](<a href=“http://www.wolframalpha.com/]Wolfram|Alpha:”>http://www.wolframalpha.com/)</p>

<p>Your bestfriend online</p>

<p><a href=“f(x,y)''=√(x^2+y^2)[/url] - Wolfram|Alpha”>f(x,y)''=√(x^2+y^2) - Wolfram|Alpha;

<p>f(x, y) = x c<em>2(y)+c</em>1(y)+1/6 ((x^2-2 y^2) sqrt(x^2+y^2)+3 x y^2 log(sqrt(x^2+y^2)+x))</p>

7

<p>Wait how did you guys know its a minimum? I got yhat the critical point is (0,0) but why is it a min?</p>

8

<p>As the others above said the value of f(x,) at (0,0) is 0, the value at all other points is > 0. therefore f(x,y) must increase as you move away in any direction from the point (0,0) => it’s a minimum.</p>

9

<p>Gotcha. I see. You guys are awesomeness! Thanks.</p>