Question

Find all local extrema for the function f(x,y) = -6x? -5xy - 3y2 + 111x +58 +5. Select the correct choice below and fill in any answer boxes within your choice. O A. There are local maxima located at (Simplify your answers. Type ordered pairs. Use a comma to separate answers as needed.) O B. There are no local maxima. Select the correct choice below and fill in any answer boxes within your choice. O A. There are local minima located at (Simplify your answers. Type ordered pairs. Use a comma to separate answers as needed.) O B. There are no local minima Select the correct choice below and fill in any answer boxes within your choice. O A. There are saddle points located at (Simplify your answers. Type ordered pairs. Use a comma to separate answers as needed.) OB. There are no saddle points

Answer 1

fy f(x, y) = -6x² – 5xy – 3y2 +111x + 58y+5 Find the partial derivative as a fa -(-6x² – 5xy+111x – 3y² + 58y+5) =-12x – 5y+111 ax a –6x’ – 5xy+111x – 3y2 + 58y+5) = -5x –6y+58 Oy Now to find critical point, solve 111-12x -12x - 5y+111=0=5y = 111-12x=y=- 5 -5s-6y+58 = 0 111-12x -58-6 + 58 = 0 5 3-666 +47 x= = -290 47x = 376 >> x=8 111-12x Substitute x = = 8 in y= 5 111-12(8) y= = 3 5 -12 -5 Find the discriminant as D= = 47>0 and fix = -12<0, hence maximum -5 -6 A. There are local maxima located at(8,3) B. There are no local minima B. There are no saddle points