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Chapter 12: Q 49. (page 976)

Find the relative maxima, relative minima, and saddle points for the given functions. Determine whether the function has an absolute maximum or absolute minimum as well.

f(x,y)=x2y2f(x, y)=x^{2} y^{2}

Short Answer

Expert verified

Critical points are (x,0),(0,y)and local minima is(x,0),(0,y).

Step by step solution

01

Given Information

It is given thatf(x,y)=x2y2

02

Finding critical points

The gradient of function is

鈭�f(x,y,z)=鈭�f鈭�xi+鈭�f鈭�yj

=2xy2i+2x2yj

The gradient vanishes at critical points 鈭�f(x,y)=0

鈬�2xy2=0and 2x2y=0

The above equation is satisfied at point (x,0),(0,y)

Hence critical points are(x,0),(0,y)

03

Calculating local minima

The second derivative is 鈭�2f鈭�x2=2y2,鈭�2f鈭�y2=2x2,鈭�2f鈭�y鈭�x=4xy

The discriminate is Hf(x,y)=鈭�2f鈭�x2鈭�2f鈭�x2-鈭�2f鈭�y鈭�x2

=2y22x2-(4xy)2

=-12x2y2

At (x,0),(0,y),Hf=0, discriminate is of no use

As f(x,y)=x2y2, so(x,0),(0,y)is local minima

At other points,f(x,y)>0

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Evaluate the following limits, or explain why the limit does not exist.

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