fsolve optimization that include corner solutions

Hello, I am working on the optimization problem via fsolve function.

I input the K-T condition that allows for the possibility of corner solutions but fail to get the expecter result.

It only gives the interior solution. how to fix it?

below are my coding:

function [c_1,c_2,lambda,miu_1,miu_2] = q3_f(p_1,p_2,a,rho,I,X0)
options = optimset('Display', 'off');      
fsolve(@cFOCs_f,X0,options);                
   function F = cFOCs_f(X)                 
          c_1       = X(1);      
          c_2       = X(2);               
          lambda    = X(3);   
          miu_1     = X(4)
          miu_2     = X(5)
          F(1)    = 1/rho *(a*c_1^rho+(1-a)*c_2^rho)^(1/rho-1)*a*rho*c_1^(rho-1)-lambda*p_1
          F(2)    = 1/rho *(a*c_1^rho+(1-a)*c_2^rho)^(1/rho-1)*(1-a)*rho*c_2^(rho-1)-lambda*p_2
          F(3)    = I-p_1*c_1-p_2*c_2
          F(4)    = miu_1*c_1
          F(5)    = miu_2*c_2
   end
end

clear; close all; clc;
 
a=0.5;
rho=2;
I=10;
p_1=1;
p_2=2;
X0=[-2,-4,-7,1,1];
[c_1,c_2,lambda,miu_1,miu_2] = q3_f(p_1,p_2,a,rho,I,X0)

[Alex Sha]

Are those the results you want?

c_1: 2
c_2: 4
lambda: 0.316227766016838
miu_1: -4.92547533265953E-161
miu_2: 2.17114164860151E-161