Please answer number 12! **Mathematics Problems for Advanced Calculus Students****Problem 11:**Find an equation of the tangent plane to the surface $\sqrt{x} + \sqrt{y} + \sqrt{z} = 4$ at the point $P(4, 1, 1)$.**Problem 12:**Find the local maximum and minimum values and saddle point(s) of the function $f(x, y) = -x^4 + 4xy - 2y^2 + 1$.**Problem 13:**Find the maximum value of the function $f(x, y) = 6 - 4x^2 - y^2$ subject to the constraint $4x + y = 5$.

Name: Please answer number 12! **Mathematics Problems for Advanced Calculus
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Description: Please answer number 12! **Mathematics Problems for Advanced Calculus Students****Problem 11:**Find an equation of the tangent plane to the surface $\sqrt{x} + \sqrt{y} + \sqrt{z} = 4$ at the point $P(4, 1, 1)$.**Problem 12:**Find the local maxim

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!

**Mathematics Problems for Advanced Calculus Students**

**Problem 11:**
Find an equation of the tangent plane to the surface $\sqrt{x} + \sqrt{y} + \sqrt{z} = 4$ at the point $P(4, 1, 1)$.

**Problem 12:**
Find the local maximum and minimum values and saddle point(s) of the function $f(x, y) = -x^4 + 4xy - 2y^2 + 1$.

**Problem 13:**
Find the maximum value of the function $f(x, y) = 6 - 4x^2 - y^2$ 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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