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Chapter 12: Q. 19 (page 989)

Partial derivatives: Find all first- and second-order partial derivatives for the following functions:

Short Answer

Expert verified

鈭�2f鈭�x2=2x3+6x2y-6xy2-2y3(x2+y2)2and鈭�2f鈭�y2=2x3-6x2y+6xy2-2y3(x2+y2)3

Step by step solution

01

Step 1. Given information

02

Step 2. Differentiate

Partially differentiating w.r.t x and y,

Now,

and,

Therefore,

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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.

Achainruleforfunctionsoftwovariables:Letf(x,y)=x2y3,x=scost,andy=ssint.Find鈭�y鈭�sand鈭�y鈭�t.

In Exercises 37-42, use the partial derivatives of gx,y=xcosyand the point 1,蟺specified to

afind the equation of the line tangent to the surface defined by the function in the xdirection,

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cfind the equation of the plane containing the lines you found in parts aand b.

In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.

f(x,y,z)=x+y+zwhenx2+4y2+16z2=64

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.

Show that the only point given by the method of Lagrange multipliers for the function f(x,y)=x2ysubject to the constraint x+y=0is(0,0).
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