Find the all the critical points of f(x, y) = 4xy - x^4 - y^4 +4 and determine t

Question

Find the all the critical points of f(x, y) = 4xy - x^4 - y^4 +4 and determine the nature of each critical point. Find the maximum value of f along the path x - y = 0. A heat seeking particle is located at the point (1, 1) on a metal plate whose temperature at (x, y) is T(x, y) = 10 - x^2 - 2y^2. Find the path of the particle as it continuously moves in the direction of maximum temperature increase. Evaluate integral_C (squareroot x^2 + 1 - x^2y) dx + (xy^2 - y^5/3) dy, where C is a path from (0, 0) to (1, 1) along the curve y = x^and from (1, 1) to (0, 0) along a straight line. Find the equation of the tangent plane to the surface x^2 + y^2 + z^2 = 5 at the point (0, 1, 2). Find the equation of the tangent line to the curve of intersection of the surfaces x^2 + y^2 + z^2 = 5 and x^2 + y^2 + z = 3 at the point (0, 1, 2). Find the area of the surface of the portion of the plane z = x inside the cylinder x^2 + y^2 - x = 0.

Explanation / Answer

1)given f(x,y)=4xy -x4-y4 +4

fx=4y -4x3-0 +0 ,fy=4x -0-4y3+0

fx=4y -4x3 ,fy=4x-4y3

for critical points fx =0, fy =0

4y -4x3=0,4x-4y3=0

y=x3, x=y3

y=(y3)3

y=y9

y(y8-1)=0

y =0, y =-1, y=1

y =0 ,x=y3 =>x=0

y =1 ,x=y3 =>x=1

y =-1 ,x=y3 =>x=-1

critical points are (-1,-1),(0,0),(1,1)

fx=4y -4x3 ,fy=4x-4y3

fxx=-12x2 ,fyy=-12y2,fxy=4,D=fxxfyy-fxy2

at (0,0)

fxx=0 ,fyy=0,fxy=4,D=-16 <0

saddle point at (0,0)

at (-1,-1)

fxx=-12 ,fyy=-12,fxy=4,D=128

local maximum at (-1,-1)

fxx=12 ,fyy=12,fxy=4,D=128

local maximum at (1,1)

on path x-y =0

=> y=x

f(x,y)=4xy -x4-y4 +4

f(x)=4xx -x4-x4 +4

f(x)=4x2 -2x4 +4

f '(x)=8x-8x3

for critical point f '(x) =0

8x-8x3=0

x=0,-1,1

f(0)=4

f(-1)=4 -2 +4 =6

f(1)=4 -2 +4 =6

maximum value along the path x-y =0 is 6

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.