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Chapter 12: Q 51. (page 932)

Determine the domains of the functions in Exercises 47鈥�56, and find where the functions are continuous.

f(x,y,z)=sin(x+y+z)x+y+z

Short Answer

Expert verified

The function f(x,y,z)=sin(x+y+z)x+y+z is continuous over its entire domain.

Step by step solution

01

Step 1. Given information. 

We have given expression:f(x,y,z)=sin(x+y+z)x+y+z

02

 To find continuous of the function.   

The given function is a rational expression of sine function and a radical function.

The domain of a sine function is the entire set of real numbers.

The radical term in the denominator makes sure that the term inside the radical sign is greater than or equal to 0.

That is x+y+z鈮�0

The radical expression means the denominator is not equal to 0.

x+y+z鈮�0x+y+z鈮�0

The domain of the given function:

Domain(f)=x,y,z:x+y+z>0

A rational expression is said to be continuous over its domain always.

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Most popular questions from this chapter

In Example 4 we found that the function f(x,y)=x3+9x2+6y2has stationary points at (0,0)and (-6,0).

(a) Use the second-derivative test to show that \(f\) has a saddle point at(-6,0).

(b) Use the second-derivative test to show that \(f\) has a relative minimum at(0,0).

(c) Use the value of \(f(-10,0)\) to argue that \(f\) has a relative minimum at (0,0),and not an absolute minimum, without using the second-derivative test.

Explain how you could use the method of Lagrange multipliers to find the extrema of a function of two variables, f(x,y),subject to the constraint that (x,y)is a point on the boundary of a triangle Tin the xy-plane.

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

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

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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