How to solve the following multivariable optimization? I found a question in the CSIR NET exam which was an optimization problem.We have to find maximum and minimum value of $f(x_1,x_2,x_3,x_4)=30x_1+90x_2+100x_3+120x_4$ subject to the constraints $x_1,x_2\geq 1,x_3,x_4\geq 1/2$ and $x_1+x_2+x_3+x_4=5$.I have done Linear programming problems but this seems to be a multivariable linear programming.Can someone give me an idea how to solve it?

The solution $x=(3,1,0.5,0.5)$ obtained via inspection by @callculus42 is indeed optimal, with objective value $30\cdot3+90\cdot1+100\cdot0.5+120\cdot0.5 = 290$.  For a short certificate of optimality (from LP duality), note that
\begin{align}
30 x_1 + 90 x_2 + 100 x_3 + 120 x_4
&= 30 \sum_{j=1}^4 x_j + 60 x_2 + 70 x_3 + 90 x_4 \\
&\ge 30 \cdot 5 + 60\cdot 1 + 70\cdot0.5 + 90\cdot0.5 \\
&= 290.
\end{align}
