Math  /  Calculus

Questiong(x,y)=x2y22x;Rg(x, y)=x^{2}-y^{2}-2 x ; R : is the closed region in the xyx y-plane bounded by the graphs y=x2y=x^{2} and y=4y=4. Absolute minimum of -17 at (1,4)(1,4); Absolute maximum of 2 at (1,1)(-1,1)

Studdy Solution
Compare values to determine absolute minimum and maximum.
Evaluate g(x,y) g(x, y) at the boundary points and critical points found:
1. On y=x2 y = x^2 for x=2,0,2 x = -2, 0, 2 : - g(2,4)=(2)2422(2)=416+4=8 g(-2, 4) = (-2)^2 - 4^2 - 2(-2) = 4 - 16 + 4 = -8 - g(0,0)=02022(0)=0 g(0, 0) = 0^2 - 0^2 - 2(0) = 0 - g(2,4)=22422(2)=4164=16 g(2, 4) = 2^2 - 4^2 - 2(2) = 4 - 16 - 4 = -16
2. On y=4 y = 4 for x=2,0,2 x = -2, 0, 2 : - g(2,4)=(2)2422(2)=416+4=8 g(-2, 4) = (-2)^2 - 4^2 - 2(-2) = 4 - 16 + 4 = -8 - g(0,4)=02422(0)=16 g(0, 4) = 0^2 - 4^2 - 2(0) = -16 - g(2,4)=22422(2)=4164=16 g(2, 4) = 2^2 - 4^2 - 2(2) = 4 - 16 - 4 = -16
The absolute minimum value is 17-17 at (1,4)(1, 4) and the absolute maximum value is 22 at (1,1)(-1, 1).
The absolute minimum value is 17-17 at (1,4)(1, 4) and the absolute maximum value is 22 at (1,1)(-1, 1).

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