Are you using the Matlab editor? If so, the orange marks indicate problems with syntax that are causing your problems. For example, hover your mouse over the double equals sign in
I==1.06*x1+1.08*x2+1.1*x3+1.12*x4+1.15*x5
and you will see a message like "Possible inappropriate use of == operator. Use = if assignment is intended." Hover it over the variable x2 in [x1 x2 ...] and you will see "Best practice is to separate output variables with commas." Fix all the highlighted problems and you get:
syms I x1 x2 x3 x4 x5 g1 g2 lambda1 lambda2
I=1.06*x1+1.08*x2+1.1*x3+1.12*x4+1.15*x5;
g1=x1+x2+x3+x4+x5;
g2=x2+1.5*x3+2.5*x4+5*x5;
[x1, x2, x3, x4, x5, lambda1, lambda2]=solve(diff(I,x1)-lambda1==0,diff(I,x2)-lambda1-lambda2==0,diff(I,x3)-lambda1-1.5*lambda2==0,diff(I,x4)-lambda1-2.5*lambda2==0,diff(I,x5)-lambda1-5*lambda2==0,g1==250000,g2==500000,x1,x2,x3,x4,x5,lambda1,lambda2);
For which you get the response, "Warning: Explicit solution could not be found." Oh, well.
NOTE ADDED: I see why you don't get any solutions. The first five equations in the solve command evaluate to:
53/50 - lambda1 = 0
27/25 - lambda2 - lambda1 = 0
11/10 - (3*lambda2)/2 - lambda1 = 0
8/25 - (5*lambda2)/2 - lambda1 = 0
23/20 - 5*lambda2 - lambda1 = 0
Five incompatible equations in two unknowns.
