i have a 2 variable function whose maxima and minima has to be found.?

Question

the first derivative test yields complex roots.. how to proceed...
the function is
f(x,y) = (x^3) + (3*x*x*y) -15 (x*x) -15 (y*y) +72x

Answer 1

f(x,y) = (x^3) + (3*x*x*y) -15 (x*x) -15 (y*y) +72x

∂f/∂x = 3x^2 + 6 xy - 30x + 72 = 0
∂f/∂y = 3x^2 - 30y = 0

since y = 3x^2/30 = x^2/10

3x^2 + 3x^3/5 - 30 x + 72 = 0 multiply by 5/3 to get
x^3 + 5x^2 - 50x + 120 = 0

I believe you are correct that this has no real roots. In that case, the function is monotonic and has no extreme points.

Answer 2

I did not do the calculations but a 3rd degree polynom with real coefficients has ALWAYS at least one real root.
df/dx= 3x^2 +6xy -30x+72=0
df/dy= 3x^2-30y=0 so y =x^2/10
3x^2+3x^3/5 -30x+72 =0 is really a 3rd degree equation