Minimum Value

Ans) f_x = 2x + 2 y + 6
f_y = 4y + 2x + 2
fx = 0 implies (x+y+3=0) .....................(i)
and f y = 0 implies (2y+x+1 = 0) .....................(ii)
On solving (i) and (ii), we get y = 2 and x = -5
(a,b) : (-5,2)

Now, let r = f xx = 2
and, s = f xy = 2
and, t = fyy = 4

Now since, rt - s² = 8 - 4 = 4 >0 and r > 0 , then the function has a min value at (a,b)

Therefore, Min value = 25 +8 - 20 -30 +4 = - 13