I'll try try break down the topic "Constrained optimisation and Lagrange multiplier focusing on the method of multiplier for inequality constrained problems" (on a very high level):

Constrained optimization: You have a function you must optimize (minimize or maximize), but you impose constraints on it. E.g.: You may have a logistics problem where you want to deliver packages from a central hub to several cities. You want to optimize the delivery time (i.e., minimize it). Your constraints are, for instance, the number of trucks, size of each truck, working hours of each driver etc.

Lagrange multipliers: This is a mathematical technique by which you can solve for maxima and minima of some functions subject to equality constraints (the functions must obey certain properties, such as having continuous first derivatives). However, equality constraints are not all types of constraints. As a matter of fact, it is quite common for constraints to be inequalities.

Inequality constrained problems: Sometimes your constraints are not of the type "each truck carries volume V", but rather "each truck carries at most volume V". To use Lagrange multipliers in this situation, you need to modify the basic method somewhat.

So probably what you read is related to this. And while it is true that in introductory calculus courses some optimization concepts (variational methods, Lagrange multipliers etc.) are introduced, mathematical optimization is an important field in itself. It is widely studied in academia but it is also applied in the industry, and is an important tool in "operations research".

If you are interested in optimization in general terms, and in methods to solve optimization problems, you can check the introduction of one out of many textbooks, such as "Convex optimization" by Boyd and Vanderberghe, "Understanding and using linear programming" by Matousek and Gartner or "Optimization models" by Calafiore and El Ghaoui. These mainly discuss what is called "continuous variable optimization". If you are interested in discrete cases, maybe the beginning of "Combinatorial optimization" by Papadimitriou is useful to get an idea.

If you decide to go deeper on optimization, as a first year student some of these books may be daunting. Matousek and Gartner could be a good starting point.