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Chapter 12: Q 52. (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.

g(x,y)=x3y

Short Answer

Expert verified

Critical points are (0,y)

There is saddle point at(0,y)

Step by step solution

01

Given Information

The given function is

g(x,y)=x3y

02

Finding critical points

The gradient of given function is

鈭�g(x,y,z)=鈭�g鈭�xi+鈭�g鈭�yj

=3x2yi+x3j

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

鈬�3x2y=0,x3=0

The critical points are(0,y)

03

Finding saddle points

The second order derivative of given function is given by

鈭�2g鈭�x2=6xy,鈭�2g鈭�y2=0,鈭�2g鈭�y鈭�x=6x

Hence, discriminate is given by Hg(x,y)=鈭�2g鈭�x2鈭�2g鈭�x2-鈭�2g鈭�y鈭�x2

=6xy(0)-(6x)2

=-36x2

As (0,y),Hg=0, discriminate is of no use.

For point 0,y, there is a saddle point as function g(x,y)=x3ychanges sign in neighbor of0,y

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Most popular questions from this chapter

Letz=f(x,y)g(x,y).Showthat鈭�z鈭�x=鈭�f鈭�xg(x,y)+f(x,y)鈭�g鈭�xand鈭�z鈭�y=鈭�f鈭�yg(x,y)+f(x,y)鈭�g鈭�y

Let f(x,y)be a differentiable function such that 鈭�f(x,y)鈮�0for every point in the domain of f, and let Rbe a closed, bounded subset of role="math" localid="1649887954022" 鈩�2.Explain why the maximum and minimum of f restricted to Roccur on the boundary ofrole="math" localid="1649888770915" R.

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Solve the exact differential equations in Exercises 63鈥�66.12x3y5鈭�15x4y6dydx=0.
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