Introduction to Maxima and Minima
Maxima and minima are two important concepts of calculus which are used to find the critical points of a function. These points are calculated by finding the derivative of a function. It is used to find the points where a function takes maximum and minimum values. Let’s understand more about maxima and minima. Also understand how to find maxima and minima of a function.
Understanding of the Maxima and Minima
In mathematical analysis, the maxima and minima are the highest and lowest values that a function takes either within the given domain or the whole domain. Moreover, these values are also referred to as the extrema of a function. This concept is important in optimization problems where we need to calculate the maximum and minimum values of a function.
Definition of Maxima and Minima
A function f(x) has a maximum value at a point x=c if the value of the function at x=c is greater than or equal to f(x) for all points, mathematically,
\[ f(c) \geq f(x)^2 \]
Whereas the function f(x) has a minimum value at x=c if the value of the function at x=c is less than or equal to f(x) for all points. Mathematically,
\[ f(c) \leq f(x)^2 \]
These points of a function also play an important role in continuity of a function.
Types of Maxima and Minima
There are two main types of maxima and minima, namely,
- Absolute maxima and minima
- Relative maxima and minima
Let’s discuss these types of extremes one-by-one.
Absolute Maxima and Minima
The point at which a function takes the highest value within the entire domain is known as absolute maxima. Similarly, the point at which the function takes the lowest value within the entire domain is called absolute minima. The absolute maxima and minima can also be referred to as the global maxima and minima. Remember that there can only be one absolute maxima and minima of a function.
Relative Maxima and Minima
The relative maxima and minima of a function or a curve are also known as the local maxima and minima. The local maxima is the value of a function f(x) at a point x=c which is always greater than the values of the function at the neighboring points i.e. f(c)>f(x). Whereas the local minima of relative minima is the value of a function f(x) at a point x=c which is less than the values of the function at the neighboring points i.e. f(c)<f(x).
How do you calculate the maxima and minima of a function?
The maxima and minima of a function can be calculated by using derivatives. Here are some simple steps to calculate extreme points of a function.
- Differentiate the given function f(x).
- Equate the derivative of f(x) to zero i.e. f’(x)=0 and solve it to calculate the value of x. It will be the critical point of the function.
- Now calculate the second derivative of f(x) and calculate the value of f’’(x) at the critical point calculated in step 2.
- Now analyze the values of f’’(x). If f’’(x)>0 then the graph of f(x) is concave up and the function has a local minimum at that point.
- If f’’(x)<0 then the graph of the function is concave down and the function has a local maximum at that point.
Let’s understand how to calculate maxima and minima of a function in the following example.
Maxima and minima example
Find the maxima and minima of the function,
\[ f(x) = x^3 + x^2 \]
First, we will calculate the derivative of the function. Since the function contains an algebraic function with exponents, therefore we will use the power rule of the derivative.
\[ f'(x) = 3x^2 + 2x \]
Now equating f’(x)=0,
\[ 3x^2 + 2x = 0 \]
\[ x(3x + 2) = 0 \]
\[ x = 0, \quad x = -\frac{2}{3} \]
Now, using these values in f(x), we get,
f(0)=0
f(-⅔)=4/27
Hence the stationary points are (0, 0) and (-⅔, 4/27) where (0, 0) is the local minimum of f(x) and (-⅔, 4/27) is the local maximum.
Conclusion
Maxima and minima are two important concepts in calculus which are used to optimize a function. It has many applications in mathematical analysis as well as in other fields of science. For example, it helps to optimize the rate of profit and loss at some specific factors. The maxima and minima of a function are also known as its extreme points.