Find the local extreme values of the function f(x,y) = xy - x2 - y2 - | Basic Mathematics 0

Basic Mathematics05 marksNumerical: local extreme valuesLocal extrema of multivariable functions

Find the local extreme values of the function $f(x,y) = xy - x^2 - y^2 - 2x - 2y + 4$ . [5]

Find the local extreme values of the function $f(x,y) = xy - x^2 - y^2 - 2x - 2y + 4$. [5]

Function: $f(x,y) = xy - x^2 - y^2 - 2x - 2y + 4$ First partial derivatives: $fx = y - 2x - 2$ $fy = x - 2y - 2$ Set equal to zero: $y - 2x - 2 = 0 \quad (1)$ $x - 2y - 2 = 0 \quad (2)$ From (1): $y = 2x + 2$ Substitute into (2): $x - 2(2x+2) - 2 = 0$ $x - 4x - 4 - 2 = 0$ $$-3x - 6 = ...

Function: $f(x,y) = xy - x^2 - y^2 - 2x - 2y + 4$ First partial derivatives: $$fx = y - 2x - 2$$ $$fy = x - 2y - 2$$ Set equal to zero: $$y - 2x - 2 = 0 \quad (1)$$ $$x - 2y - 2 = 0 \quad (2)$$ From (1): $y = 2x + 2$ Substitute into (2): $$x - 2(2x+2) - 2 = 0$$ $$x - 4x - 4 - 2 = 0$$ $$-3x - 6 = ...

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Find the local extreme values of the function f(x,y) = xy - x2 - y2 - 2x - 2y + 4.