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Chapter 12: Q 51. (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)=x3y2

Short Answer

Expert verified

Critical points are (x,0),(0,y)

Local minima is at (x,0)withx>0

Local maxima is at (x,0)withx<0

Saddle point is at(0,y)

Step by step solution

01

Given Information

The given function isf(x,y)=x3y2

02

Finding critical points

The gradient of function is given by

鈭�f(x,y,z)=鈭�f鈭�xi+鈭�f鈭�yj

=3x2y2i+2x3yj

Gradient vanishes at 鈭�f'(x,y)=0

鈬�3x2y2=0,2x3y=0

Hence critical point is(x,0),(0,y)

03

Finding second derivative

The second order derivative of function are

鈭�2f鈭�x2=6xy2,鈭�2f鈭�y2=2x3,鈭�2f鈭�y鈭�x=6x2y

Hence, discriminate is

Hf(x,y)=鈭�2f鈭�x2鈭�2f鈭�x2-鈭�2f鈭�y鈭�x2

=6xy22x3-6x2y2

=-24x4y2

04

Determining maxima, minima and saddle point

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

For f(x,y)=x3y2

  • For (x,0)withx>0, there will be local minima as f(x,y)>0for all other point in neighbor of (x,0)
05

Explanation

  • For (x,0)withx<0, there is local maxima.
  • For (0,y), there is saddle point for the function asf(x,y)>0ifx>0andf(x,y)<0ifx<0in neighbor of(0,y)

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Most popular questions from this chapter

Fill in the blanks to complete the limit rules. You may assume that limx鈫�a鈥�f(x)andlimx鈫�a鈥�g(x) exists and that k is a scalar.

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Describe the meanings of each of the following mathematical expressions:

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