How do i find the "local extrema" for a function like this: f(x,y) = xy+y-15x

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- Nov 25th 2009, 05:43 AMsquareLocal Extrema help
How do i find the "local extrema" for a function like this: f(x,y) = xy+y-15x

- Nov 25th 2009, 06:04 AMSoroban
Hello, square!

Quote:

How do i find the "local extrema" for a function like this: .$\displaystyle f(x,y) \:=\: xy+y-15x$

And test the critical value(s) in: .$\displaystyle D \;=\;\left(\frac{\partial^2\!f}{\partial x^2}\right)\left(\frac{\partial^2\!f}{\partial y^2}\right) - \left(\frac{\partial^2\!f}{\partial x\partial y}\right)^2$

If all this is meaningless to you,

. . (1) you should not have been assigned this problem, or

. . (2) you weren't paying attention in class.

- Nov 25th 2009, 06:15 AMtonio

Solve the equations $\displaystyle f_x=y-15=0\,,\,\,f_y=x+1=0$ , and then use the Hessian's determinant in the points $\displaystyle (x,y)$ that you found above,$\displaystyle f_{xx}f_{yy}-(f_{xy})^2$ : if this determinant is positive and (1) $\displaystyle f_{xx}>0$ , then the point you found is a local minimum, and if (2) $\displaystyle f_{xx}<0$ the point is then a local maximum .

If the determinant is negative then the point is a saddle point (not max. not min.), and if the determinant is zero then the Hessian matrix's test fails to decide.

Tonio