Maxima and Minima

Everything in the world has its upper limit, referred to as the maximum of that thing, and the lower limit, depicting the minimum of something. If you have eight dollars with you then you can ultimately spend either eight at most or lower than that too. 

Here, eight dollars is the extreme spending expense that you can go for, which is referred to as the maximum value. On the other hand, in the case of minimum value, let’s take an instance where you are required to be at least 18 years of age to be eligible for voting. Thus, 18 here is the minimum value of age that one can go for voting. 

In mathematics, two similar concepts are known by the terms, maxima, and minima. Together, maxima and minima are referred to as the maximum and minimum value of the function or the extreme value that a function holds. 

What are maxima and minima in mathematics?

In mathematics, maxima and minima are defined as the absolute max and min values of a function. These values are specified under the given set of ranges of the function. Diagrammatically, one can comprehend this as the highlands and dells of the curve of the function. 

Interestingly, mathematics has other ways of deducing the maximum and minimum values without even seeing what the graph looks like for a function. Here, we may find two types of minima and maxima of a function, one is the local minima and maxima and the other one is the absolute maxima and minima. 

Types of maxima and minima

Local minima and maxima 

These are the maxima and minima values of a function that lies between a particular expanse. Specifying the interval to the function, local maxima refers to the value where the function is the utmost and all the values that lie near to that value are always less than the local maxima value. On the other hand, the local minima value refers to the least value of a function where the values that are near that value are always greater than this local minima value. 

Absolute maxima and minima 

The absolute maxima and minima take into consideration the entire domain of the function and not just a particular interval. Absolute maxima or the global maxima refers to the maximum value that a function can have, irrespective of the range thresholds. On the other hand, the absolute minima or the global minima refers to the minimum value that a function can have, irrespective of the range thresholds. 

How to find of a real-valued function?

Before working on finding Maxima and Minima of a real-valued function, one should have quite decent knowledge about the derivatives and how one can perform derivative tests of a function. For the case of maxima and minima, the maximum and minimum values can be found without even having a glance at the graph, by just the first-order and second-order derivative test. 

Steps to find out maxima and minima with an example

Suppose if there is a function, 

                   3x³ – 2x² + 7 = 0

Step 1: Initially, we will apply the first-order derivative test on the above equation and differentiate both sides of the equation. 

         dy/dx = d( 3x³ – 2x² + 7 )/dx

dy/dx = d (3x³)/dx- d(2x²)/dx + d(7)/dx

               dy/dx = 9x² – 4x + 0

                  dy/dx = 9x² – 4x

Step 2: To find the critical points for the function, we will keep what we have got from the first-order derivative test equal to zero. 

                        9x² – 4x = 0

                        x(9x-4) = 0

       Here, x is either equal to 0 or 4/9

Step 3: We will now again differentiate the outcome that we got in step 1 after the first-order derivative test. 

         d²y/dx² = d(9x² – 4x)/dx

       d²y/dx² = d(9x²)/dx – d(4x)/dx

                d²y/dx² = 18x – 4

Step 4: To know which critical point is the minima and which point is the maxima, we will put the critical values that we found in step 2 in the derivative we got in step 3. 

     At x = 0,   d²y/dx² = 18(0)- 4

                          -4 < 0

Thus, x=0 is the point where we will be getting the maxima of the function.

    At x = 4/9,  d²y/dx² = 18(4/9) – 4

                            4 > 0

Thus, x = 4/9 is the point where we will be getting the minima of the function. 

Outcome: From the above procedure we get that the maxima and minima points of the function 3x³ – 2x² + 7 = 0 are at x = 0 and 4/9 respectively. 

Conclusion 

In understanding, Maxima refers to the absolute utmost value that you can have of a function. On the other hand, Minima refers to the least value that you can get from a real-valued function. On comprehending the diagram, one can comprehend them as the highlands and dells of the curve of the function. The maximum and minimum values of the function can be found without even having a glimpse at the graph, simply by the first-order derivative test and the second-order derivative test.