I am a maths major, and am taking an introduction to microeconomics course this semester, and am confused by how we deduce the shape of PPF's. 

For example, I was given the following problem:

Larry, Moe, and Curly all work 10 hours in a day, and can work on either mowing lawns, or washing cars. In one hour Larry can wash one car or mow one lawn, Moe can mow one lawn, or wash two cars, and Curly can wash one cars or mow two lawns.

The problem then asked me to find the various productions of each service given how much time each person spent doing one of the jobs. From there, it asked me to draw a PPF and label my points on it.

I would like to be able to go about this from a more mathematical point of view. Let $l,m,c$ be the amount of hours Larry, Moe, and Curly spend mowing lawns respectively. Then the total production of this service in terms of $l,m,c$ is given by:
$$L(l,m,c)=l+c+2m$$
while the total production of washing cars is then:
$$C(l,m,c)=(10-l)+2(10-c)+(10-m)$$ 
The PPF is then a map:
$$\begin{align}
PPF:\mathbb{R}^3&\longrightarrow \mathbb{R}^2\\
(l,m,c)&\longmapsto (C,L)
\end{align}$$
The total production output I think is given by $L+C$, so the line we draw representing the boundary of the $PPF$ should be the maximum of $L+C$ subject to the constraint:
$$\begin{align}
0\leq l+m+c\leq 30
\end{align}$$
but proceeding with Lagrange multipliers yields a set of inconsistent equations, indicating to me that my reasoning is off somewhere but I am not sure where. Furthermore, all of these equations are linear or affine so I am not sure how I'm supposed to get a bowed out boundary of the PPF. Any help on how to think about problems like this from a mathematical stand point would be very helpful.

edit: I guess it should really be a map from $[0,10]\times[0,10]\times[0,10]\rightarrow \mathbb{R}^2$.