math about to have a test over this please help!

Unfortunately that is the entire problem..
Find and classify all local minima, local max, and saddle points of the function F(x,y)= -3x^2y-3xy^2+36xy

There should be 3 saddle points and a maxima for sure! Please help me get the points. I can't do these maxima minima problems. they are confusing as heck! Im about to have a test over this, freaking out!!

  1. 0
  2. 0
  3. 16
asked by Lucy
  1. F(x,y)=-3x^2y-3xy^2+36xy
    For maximum/minimum,Fx=0 AND Fy=0
    Fx=36y-6xy-3y^2=3y(12-2x-y)=0 ...(1)
    Fy=36x-6xy-3x^2=3x(12-2y-x)=0 ...(2)

    Clearly (0,0) is a critical point.
    If y=0, solve for x in (2) to get x=12, so
    (12,0) is another critical point.
    Similarly (0,12) is another critical point.
    Finally, solve for x,y in
    12-2x-y=0 and
    12-x-2y=0 to get
    x=y=4, or (4,4) is a critical point.

    So the four critical points are
    (0,0)
    (0,12)
    (12,0)
    (4,4)

    Second derivative test
    Calculate
    Fxx=-6y
    Fyy=-6x
    Fxy=Fyx=6(6-x-y) [Clairot's theorme]
    Now calculate
    D(x,y)=Fxx.Fyy-Fxy²
    =36xy-6(36-12(x+y)+(x+y)^2)
    D(0,0)=-216<0 => saddle point
    D(0,12)=-216<0 => saddle point
    D(12,0)=-216<0 => saddle point
    D(4,4)=552 >0 => max or min
    Fxx=-6(4)=-24 <0 => maximum

    1. 0
    2. 0
    posted by MathMate

Respond to this Question

First Name

Your Response

Similar Questions

  1. calculus

    find and classify all local minima, local maxima, and saddle points of the function f(x,y)= e^-y(x^2+y^2)

    asked by karen on July 13, 2010
  2. cal3

    Find and classify all local minima, local maxima, and saddle points of the function f(x,y)= -3yx^2-3xy^2+36xy

    asked by Lucy on February 27, 2012
  3. cal3

    Find and classify all local minima, local maxima, and saddle points of the function f(x,y)= -3yx^2-3xy^2+36xy I really need help answering this one!! Please answer in detail! Thank you

    asked by Lucy on February 28, 2012
  4. cal3/algebra help me!

    Find and classify all local minima, local maxima, and saddle points of the function f(x,y)= -3yx^2-3xy^2+36xy I really need help answering this one!! Please answer in detail! Thank you This is what I have so far...

    asked by Lucy on February 28, 2012
  5. Calculus (pleas help!!!)

    Let p(x)=x^2(a−x), where a is constant and a greater than 0. Find the local maxima and minima of p. (Enter your maxima and minima as comma-separated xvalue,classification pairs. For example, if you found that x=−2 was a local

    asked by Claudia on April 7, 2011
  6. Calculus - You wont get it

    Find the local max and local min and saddle points of f(x, y) = x^4 + y^4 - 4xy +1

    asked by Big Deal on May 23, 2011
  7. Math

    need to classify to stationary points as local minimum and local maximum of g(x) got this far 2x-7=0 4x+1=0 so x = 7/2 and x = 1/4 but bit confused on using xL and xR for each to get the local min and max..?

    asked by Pd on September 12, 2010
  8. How to sketch graph of...

    a differentiable function y =f(x) that has the given properties. 1. local minimum at (1,1), and local maximum at (3,3) 2. local minima at (1,1) and (3,3) 3. local maxima at (1,1) and (3,3) I don't remember how to do this. Thank

    asked by Jen on November 3, 2006
  9. basic calculus

    use the first and second derivatives to find the x-coordinates of all local maxima and minima of f(x)=8e^2x-2x. indicate which of your answers are local minima and which are local maxima.

    asked by Connor on May 14, 2012
  10. college algebra

    Let f(x)= x^5/120 - x^3/6 + x Find the local minima and maxima, then determine where the function is decreasing and increasing. Basically, I just need to know how to type this correctly in my calculator so I can get the right

    asked by Zach on April 2, 2008

More Similar Questions