## Maximizing problem with an inequality constraint.

Hello I have a worked example where I have to maximize a function with an inequality constraint. The problem is worked out below.

I know how to get the kuhn-tucker conditions in the first step. I also understand how the first two possibilities (i) and (ii) are ruled out. In the third one the value of $\lambda$ is 2. But how did the problem work out the value of and x2 to be 10?

Thanks.
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 In case (iii), x_1=0. Why not just go back to the original conditions, you are trying to maximize y=2*x_2 subject to the constraint x_2<=10, so clearly we have a max when x_2=10, thus y=20.