Find and classify the critical points of each function fxy

Solution

The critical points are when df/dx = 0 and df/dy = 0 ; (i) df/dx = 6x² + y² + 10x = 0 (ii) df/dy = 2xy + 2y = 0 ? 2y( x + 1) = 0 with solutions x = - 1 or y = 0 ; Substitute in (i) : x = -1 ? 6 + y² - 10 = 0 ? y² = 4 ? y = ± 2 ; y = 0 ? 6x² + 10x = 0 ? x = 0 or x = - 5/3 : So critical points are (-1,-2) ; (-1,2) ; (0 ,0) ; ( -5/3, 0) For determining the nature of the critical points ( x0 , y0 ), evaluate : D(x0,y0) = d²f/dx² * d²f/dy² - ( d²f/dxdy )² in the critical points , this way : if D > 0 and d²f/dx² in (x0,y0) > 0 , then f(x0,y0) is a relative minimum value ; if D > 0 and d²f/dx² in (x0,y0) < 0 , then f(x0,y0) is a relative maximum value ; if D < 0 , then f(x0,y0) is a saddle point. d²f/dx² = 12x + 10 d²f/dy² = 2x + 2 d²f/dxdy = 2y I see that max and min points are at (0 ,0) ; ( -5/3, 0) ; and saddle points at (-1,-2) ; (-1,2)