Difficulty in an economics' optimization problem using Kuhn-Tucker conditions (interpretation difficulty)
The OP clarified in a comment that a) "income" here means "revenues" and not "profit" (it is not always the case), and b) that Karush-Kuhn-Tucker multipliers work just fine if we define them to be non-positive instead of non-negative (they do, but it is not a widely known fact).  

The other bad terminology in the statement of the problem is the "unit cost" one -it is really meant "marginal cost". So we have to obtain the Total Cost function from its partial derivatives. This is easy to do since we see that the cross-partial is zero.  

So if 

$$\frac {\partial TC}{\partial x_1} = x_1,\;\;  \frac {\partial TC}{\partial x_2} = 2x_2$$ 

it follows that

$$TC = \frac 12 x_1^2 + x_2^2 + FC,\;\;\;FC\geq 0$$

and we want to minimize it subject to the constraint $p_1x_1 + p_2x_2 \geq \bar R$.  

PS: It appears that the revenue floor is $3$?