Expert: Paul Klarreich - 2/6/2011

Question
hi, i need help with this question.

The function given by f(x; y) = xy cos(xy) has a critical point at (0; 0).
Is it a local minimum, local maximum or a saddle point?

A heated metal plate P is represented by a rectangle in the xy-plane, with vertices (0; 0), (4; 0), (4; 2) and (0; 2). At each point (x; y) of P the temperature, in degrees Celsius, is given by
T(x; y) = 10x2 - 20xy + 20y + 20
Find the point(s) at which the temperature of the plate is greatest and find the greatest temperature.

thank you.

Answer
Questioner: julie
Country: Australia
Category: Calculus
Private: No
Subject: calculus.
Question: hi, i need help with this question.

The function given by f(x; y) = xy cos(xy) has a critical point at (0; 0).
Is it a local minimum, local maximum or a saddle point?

A heated metal plate P is represented by a rectangle in the xy-plane, with

vertices (0; 0), (4; 0), (4; 2) and (0; 2). At each point (x; y) of P the

temperature, in degrees Celsius, is given by
T(x; y) = 10x2 - 20xy + 20y + 20
Find the point(s) at which the temperature of the plate is greatest and find

the greatest temperature.

thank you.
......................................
Here are my computations for the first:

fx = y (cos(xy) - xy sin(xy))


fxx = y(-y sin(xy) - y( sin (xy) + xy cos (xy)))

   = -2 y^2 sin(xy) - xy^3 cos (xy)


fy = x (cos(xy) - xy sin(xy))

fyy = -2 x^2 sin(xy) - x^3y cos (xy)


fxy = cos(xy) - xy sin(xy) + y(- x sin(xy) - x sin(xy) - x^2y cos(xy))  

fxy = cos(xy) - xy sin(xy) - y(2x sin(xy) + x^2y cos(xy))  


Now you want to compute  D = fxx fyy - (fxy)^2 at (0,0)
See:

http://www.ucl.ac.uk/Mathematics/geomath/level2/pdiff/pd9.html


fxx = y(- y sin(xy) - y^2 cos(xy)) = 0
fyy = 0
fxy = 1
D = - 1

D < 0  --> neither local max or min  
......................................
To check these, try:

http://www.calculator-grapher.com/derivative-calculator.html


..................................
To do your second (which I won't):

1. Find your extreme points WITHIN the region.
2. Compute T at each of them.
3. Compute T at each of the CORNERS.

Pick the biggest.