Hi... please can someone help me or point me in the right direction as to where I can post this in order to get some help - thanks!

I am working through past exam papers and came across a question which I thought seemed easy but I'm really stuck! The question considers the following program:

Max 3*(x1)+(x2)

s.t. ((x1)-1)^3 +(x2) <= -1

(x2) <= 7

Part a) Asks to transform this to a standard form and discuss whether KKT or Lagrange optimality principle could be used to solve it.

From what I can gather from my notes, Lagrange optimality can only be used when we have equality constraints so I argued for KKT. (This may be where I went wrong!)

Part b) asks to identify all local solutions to the problem using the method of my choice for part a). Below I've written the optimality condidtions I have found:

Largrange function: L(x,s,t) = 3*(x1)+(x2) + s(((x1)-1)^3 +(x2) +1) +t((x2) - 7)

KKT conditions:

dL/d(x1) = 3 + 3*s*((x1) - 1)^2 = 0

dL/d(x2) = 1 +s +t = 0

dL/ds = ((x1)-1)^3 +(x2) + 1 <= 0

dL/dt = (x2) - 7 <= 0

s*dL/ds = s(((x1)-1)^3 +(x2) +1) = 0

t*dL/dt = t((x2) - 7) = 0

s,t >= 0

Then I went on to try and solve it but immediately I saw a major problem: From the first KKT condition, because of the squared term and the +3 on the LHS, the only way that the LHS can equal 0 is if s is negative. But this cannot happen because of the last condition. So now I am really stuck. I am wondering if perhaps I haven't applied the method right as it is only shown in my notes for minimisation problems, or I am wondering if the Lagrange optimality method was the way to go for some reason I don't understand.

Any help would be very very much appreciated. The exam is looming and I'm really confused!

Thanks!!!