what is optimization problem? what is its importance in our daily life?

Optimization problems are essentially the epitome of being some of the most useful, semi complicated math topic for our day to day lives. Say you are a writer, and you want to know what time of year is best to publish a book. You could look at a single author's publishing history (preferably someone who publishes a lot), and see what time of year they tend to publish their book, and how many copies were sold on opening day. Optimizing is looking at that data and figuring out what time of year tends to have the highest sales on opening day. Or say you are an athlete and you want to know how much sleep to get, or how much you need to warm up to perform the best during practice or a game. You can plot out how much sleep you get to how awake you feel, or how long you warm up to how well you think you did during practice.

normally after finding all of these data points you would try to find a function that fits your data. In other words you would find an equation like:

y = ax^2 + bx + c

that will pass through most of the data points you collected, and then once you find that function, you would figure out what point is the exact top of that function. Then you know you have found out how to get the optimal results from the thing you want to optimize

Basically think of any time you have thought "Man... do I really need to be doing this for so long?" or "man... is it really worth all this money?" or really any time you want to know if you need to put more resources into something, or less resources into something, optimizing can help you get closer to your answer.

The exact way to solve an optimization problem can depend on the problem. There are ways to do it which give you a really basic estimate that will probably get you close enough to what you need even just using a program like Excel. Other times if you need a really specific answer, it involves understanding calculus to get a really detailed answer.

Hope that helps!