What is the name of this method used to solve a nonlinear problem?

The lecturer taught this method in my Optimization and Control Theory Class and I wasn't quite there when he named it. Could you help me out?

He gave the following example of the method in class:

Example: Solve $$\text{max} [ f(x) = x_1 (30 - x_1) + x_2 (50 -2x_2) - 3x_1 - 5x_2 - 10x_3]$$ Subject to the constraints: $$x_1 + x_2 \le x_3 \text{and } x_3 \le 17.25$$

Solution: Begin by converting constraints to form:

$$g_1(\bar x) = x_1 + x_2 - x_3 \le 0$$ $$g_2(\bar x) = x_3 - 17.25 \le 0$$ Then: $$L(\bar x, \bar \lambda) = f(\bar x) \pm \left( \lambda_1 g_1(\bar x) + ... + \lambda_m g_m(\bar x) \right)$$

And then he proceeded as follows:

\begin{align} D_{\bar x} L: & \frac{\partial L}{\partial x_1} = 30 - 2x_1 - 3 - \lambda_1 = 0 \\ & \frac{\partial L}{\partial x_2} = 50 - 4x_2 - 5 - \lambda_1 = 0\\ & \frac{\partial L}{\partial x_3} = -10 + \lambda_1 - \lambda_2 = 0 \\ \end{align}

A system of equations is found and solved with further constraints: $\lambda_1 \ge 0$ and $\lambda_2 \ge 0$:

$$\lambda_1(x_1 + x_2 + x_3) = 0$$ $$\lambda_2 (x_3 - 17.25) = 0$$

(Through trial and error it is found that $\lambda_1 > 0$ and $\lambda_2 = 0$ is the best condition to solve this system)

Ultimately we get the solution: $$x_1 = 8.5$$ $$x_2 = 8.75$$ $$x_3 = 17.25$$

And the equation is solved.

-- So I want to read more background on this method. I'd like to know what its called. I notice the 'L' as the name of the function. Could it be Lagrange? Or something of the sort?

Yes, $L$ is called the Lagrangian. In calculus, if you want to minimize $f$, you write down the optimality condition $\nabla f(x) = 0$ and solve for $x$. But what if you want to minimize $f$ subject to certain constraints? In this situation too you can write down the optimality conditions (now called KKT conditions) and solve for $x$.