please answer number 6. Find the extrema on the disk Find the domain and range o

Question

please answer number 6. Find the extrema on the disk

Find the domain and range of the function f(x, y) = 1/x^2 + y^2 + 1 and sketch the curve f (x, y) = 1/2. Let g(x, y) = e^x + y sin(xy). Find the directional derivative of g(x, y) at point (1, pi) in the direction of v = i + j. Given equation 2xy + e^x + y - 2 = 0, find the value of dy/dx at point P(0, ln 2). Find an equation for the tangent plane and normal line of the graph of the following function z = 1/x^2 + y^2 at the point (1, 1, 1/2). Find the relative maxima, relative minima and saddle points of the function f(x, y) x^2 + y^2 - Sx - xy over the xy-plane Find extrema of f(x, y) = x^2 + y^2 - 3x - xy on the disk x^2 + y^2 lessthanorequalto 9. Find the minimum distance from the point (4, 4) to the curve x^2 + (y - 1)^2 = 4 Find the maximum value of the directional derivative of the function g(x, y) ye^-x at the point (0, 5). Given omega = sin(2x + 3y) and x = s + t, y - s - t. Find delta omega/delta omega and delta omega/delta omega

Explanation / Answer

f = x^2 + y^2 - 3x - xy

fx = partial der of f with x
fx = 2x - 3 - y = 0

fy = 2y - x = 0

So, we have :
2x - y = 3
2y - x = 0

Solving by substitution :
2(2y) - y = 3
3y = 3
y = 1

And with that x = 2

So, (2,1) which does lie within the disk....

The endpts of the disk are
(-3,0) , (3,0) , (0,3) and (0,-3)

f = x^2 + y^2 - 3x - xy

Now, we plug in all 4 points above...

f(2,1) = -3
f(-3,0) = 18
f(3,0) = 0
f(0,-3) = 9
f(0,3) = 9

So, we have :

absolute max = 18 when (-3,0)
absolute min = -3 when (2,1)

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