bharatmaheshwari96 / maxima_and_minima

What are Maxima and Minima of a Function?

Maxima and minima are the peaks and valleys in the curve of a function. There can be any number of maxima and minima for a function. In calculus, we can find the maximum and minimum value of any function without even looking at the graph of the function. Maxima will be the highest point on the curve within the given range and minima would be the lowest point on the curve.

The combination of maxima and minima is extrema. In the image given below, we can see various peaks and valleys in the graph. At x = a and at x = 0, we get maximum values of the function, and at x = b and x = c, we get minimum values of the function. All the peaks are the maxima and the valleys are the minima.

There are two types of maxima and minima that exist in a function, which are:

1. Local Maxima and Minima
2. Absolute or Global Maxima and Minima

Let us learn about them in detail.

1. Local Maxima and Minima Local maxima and minima are the maxima and minima of the function which arise in a particular interval. Local maxima would be the value of a function at a point in a particular interval for which the values of the function near that point are always less than the value of the function at that point. Whereas local minima would be the value of the function at a point where the values of the function near that point are greater than the value of the function at that point.

Local Maxima: A point x = b is a point of local maximum for f(x) if in the neighborhood of b i.e in (b−𝛿, b+𝛿) where 𝛿 can be made arbitrarily small, f(x) < f(b) for all x ∈ (b−𝛿, b+𝛿)∖{b}. This simply means that if we consider a small region (interval) around x = b, f(b) should be the maximum in that interval. Local Minima: A point x = a is a point of local minimum for f(x) if in the neighbourhood of a, i.e. in (a−𝛿,a+𝛿), (where 𝛿 can have arbitrarily small values), f(x) > f(a) for all x ∈ (a−𝛿,a+𝛿)∖{a}. This means that if we consider a small interval around x = a, f(a) should be the minimum in that interval.

2. Absolute Maxima and Minima The highest point of a function within the entire domain is known as the absolute maxima of the function whereas the lowest point of the function within the entire domain of the function, is known as the absolute minima of the function. There can only be one absolute maximum of a function and one absolute minimum of the function over the entire domain. The absolute maxima and minima of the function can also be called the global maxima and global minima of the function.

Absolute maxima: A point x = a is a point of global maximum for f(x) if f(x) ≤ f(a) for all x∈D (the domain of f(x)). Absolute minima: A point x = a is a point of global minimum for f(x) if f(x) ≥ f(a) for all x∈D (the domain of f(x)).