Please answer number 12! 11. Find an equation of the tangent plane to the surface Vr+VGtvz = 4 at the point P(4,1,1).12. Find the local maximum and minimum values and saddle point(s) of the function f(x,y) =-x4 + 4xy – 2y² + 1.%3D13. Find the maximum value of the function f(x,y) = 6 –4x2 - y? subject to the constraint4x + y = 5.

Name: Please answer number 12! 11. Find an equation of the tangent plane to
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Description: Please answer number 12! 11. Find an equation of the tangent plane to the surface Vr+VGtvz = 4 at the point P(4,1,1).12. Find the local maximum and minimum values and saddle point(s) of the function f(x,y) =-x4 + 4xy – 2y² + 1.%3D13. Find the maximum

Given ,fx,y=-x4+4xy-2y2+1fx=-4x3+4yfy=4x-4yNow,-4x3+4y=0⇒-x3+y=0…

Answered: Find the local maximum and minimum…

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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Please answer number 12!

11. Find an equation of the tangent plane to the surface Vr+VGtvz = 4 at the point P(4,1,1).
12. Find the local maximum and minimum values and saddle point(s) of the function f(x,y) =
-x4 + 4xy – 2y² + 1.
%3D
13. Find the maximum value of the function f(x,y) = 6 –4x2 - y? subject to the constraint
4x + y = 5.

Expert Solution

Step 1

$G i v e n, f_{(x, y)} = - x^{4} + 4 x y - 2 y^{2} + 1 f_{x} = - 4 x^{3} + 4 y f_{y} = 4 x - 4 y N o w, - 4 x^{3} + 4 y = 0 \Rightarrow - x^{3} + y = 0 . . . . . . (1) 4 x - 4 y = 0 \Rightarrow x - y = 0 \Rightarrow x = y$

Step 2

$F u r t h e r, s u b s t i t u t e t h e v a l u e o f y i n (1), - x^{3} + x = 0 x (- x^{2} + 1) = 0 - x^{2} + 1 = 0 a n d x = 0 x^{2} = 1 x = \pm 1 y = x = \pm 1 C r i t i c a l P o i n t s : (0, 0), (- 1, - 1), (1, 1)$

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