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Chapter 12: Q. 14 (page 989)

Find the set of points where the function is continuous.

f(x,y)=x+yx2+y2

Short Answer

Expert verified

f(x,y)=x+yx2+y2iscontinuousforall(x,y)in鈩�2exceptat(0,0).

Step by step solution

01

Step 1. Given information is: 

f(x,y)=x+yx2+y2

02

Step 2. Finding Points

Givenfunctionisrationalfunctionwithdenominatorx2+y2.Sof(x,y)iscontinuousatallpoints(x,y)in鈩�2exceptatthosepointswherex2+y2=0Since,x2+y2=0ifx=y=0Therefore,f(x,y)=x+yx2+y2iscontinuousforall(x,y)in鈩�2exceptat(0,0).

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Most popular questions from this chapter

Letz=f(x,y)g(x,y).Showthat鈭�z鈭�x=fx(x,y)g(x,y)鈭�f(x,y)gx(x,y)(g(x,y))2and鈭�z鈭�y=fy(x,y)g(x,y)鈭�f(x,y)gy(x,y)(g(x,y))2

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.

Sketch the level curves f(x, y) = c of the following functions for c = 鈭�3, 鈭�2, 鈭�1, 0, 1, 2, and 3:

f(x,y)=x鈭�sin鈭�1鈦�y

Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.

f(x,y)=tan鈭�1鈦�yx,P=(鈭�2,2)

In Exercises, by considering the function f(x,y)=x2ysubject to the constraint x+y=0,you will explore a situation in which the method of Lagrange multipliers does not provide an extremum of a function.

Why does the method of Lagrange multipliers fail with this function?
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