Question 3. The equation 2x + y? +z? + 2.xy – 2x – 2y – 6z +1 = 0 implicitly defines a function z = z(x, y) satisfying z > 3. (a) Express and as functions of x, y, z. (b) Verify that z(2, –2) = 5. Use the linearization to estimate z(2.1, -2.1).
Question 3. The equation 2x + y? +z? + 2.xy – 2x – 2y – 6z +1 = 0 implicitly defines a function z = z(x, y) satisfying z > 3. (a) Express and as functions of x, y, z. (b) Verify that z(2, –2) = 5. Use the linearization to estimate z(2.1, -2.1).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question

Transcribed Image Text:Question 3. The equation
2.x + y? + z? + 2.xy – 2x – 2y – 6z +1= 0
implicitly defines a function z =
z(x, y) satisfying z > 3.
(a)
Express and as functions of x, y, z.
(b)
Verify that z(2, –2) = 5. Use the linearization to estimate z(2.1, –2.1).
Find all critical points of z(x, y). Using the second derivative test, characterize each critical
point as a local maximum, local minimum or a saddle point.
(Hint: you will need to calculate the second order derivatives , , at each critical point. To
simplify the expression, keep in mind that and are both 0 at critical points.)
82 z
(d)
Find the maximum of z(x, y) along the line 2x +y = 5. (You don't need to prove that the
solutions you find truly give the maximum.)
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