Global Maximum

[odee]

Hi.
I need to find if this function has a global max or global min.
f(x) = ln x1 + ln x2 - x1^2 - x2^2

I know it is a strictly concave function, so any stationary point would be a global maximum
However, from the 1st partial derivatives, I arrive at:
f'(x1) = x1^-1 - 2x1
f'(x2) = x2^-1 - 2x2

If I equate it to 0, I will get + or - sq root of 0.5 for both x1 and x2!!!!

Is it possible that there can be 4 solutions of x1,x2 pairs...as against the regular 2 solutions?
Or am I not just calculating something correctly???
Also, if YES, then how do I move on from here to determine the true global maximum???

 

[Deleted member 4993]

Hi.
I need to find if this function has a global max or global min.
f(x) = ln x1 + ln x2 - x1^2 - x2^2

I know it is a strictly concave function, so any stationary point would be a global maximum
However, from the 1st partial derivatives, I arrive at:
f'(x1) = x1^-1 - 2x1
f'(x2) = x2^-1 - 2x2

If I equate it to 0, I will get + or - sq root of 0.5 for both x1 and x2!!!!

Is it possible that there can be 4 solutions of x1,x2 pairs...as against the regular 2 solutions?
Or am I not just calculating something correctly???
Also, if YES, then how do I move on from here to determine the true global maximum???

The "negative" values are extreneous - because the domain of ln(x) is (0,?). So you have only two values.

Then put those two values in your given equation.