Please SHOW ALL WORK! Multiple Choice: Only highlighted Part. Long Answer: Only

Question

Please SHOW ALL WORK!

Multiple Choice:

Only highlighted Part.

Long Answer:

Only highlighted Part.

The tangent plane to the level surface of f(x, y, z) = xyz2 at the point (2,1,1) is x + y + 2z = 5 x + 2y + 4z = 8 2x + 2y + z = 7 x + y = 3 (x - 2) + (y - 1) + (z - 1) = 0 38. Find all critical points of the functions and classify them. f(x, y) = x2y + 3y3 - y f(x, y) = x4 + 2y2 + 4xy. Find the maximum and minimum values of f(x, y, z) = xyz subject to the constraint x2 + 2y2 + 3z2 = 9. Find the volume of the given solid. Bounded by the plane 2x + 3y + z = 6 and the three coordinate planes Under the paraboloid z = x2 + y2 and above the region bounded by y = x2 and y = 4 Under the surface and above the disk x2 + y2 le 1/4 Rewrite the iterated integral and evaluate the integral. Use polar coordinates to find the volume under the paraboloid z = x2 + y2 and above the annulus in the xy-plane 4 le x2 + y2 le 16. (Sketch the region. Display the new iterated integral. Show intermediate steps.)

Explanation / Answer

f(x,y)= X^4 + 4xy+ 2y^2

fx= 4x^3+ 4y

fx =0 =4(x^3+ y) =>x^3=- y ...(i)

fy= 4x+4y

=>fy=0=4(x+y) =>x=-y ..(ii)

so

x^3-x=0 =>x=0,-1,+1

so y=0,-1,+1

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

now A=fxx=12x^2

B=fxy=4

C=fyy=1

AC-B^2= 12x^2-16

so @(0,0)=>12x^2-16=-16<0

(0,0) is saddle point

same at (-1,1) and (1,-1) =>AC-B^2=-4<0

(-1,1) and (1,-1) are saddle point

2.

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