{"id":7769,"date":"2022-04-20T09:37:57","date_gmt":"2022-04-20T04:07:57","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=7769"},"modified":"2025-06-02T20:48:23","modified_gmt":"2025-06-02T15:18:23","slug":"what-is-maxima-and-minima","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-maxima-and-minima\/","title":{"rendered":"What is maxima and minima?"},"content":{"rendered":"\n

Overview(Maxima and minima)- A function f(x) is said to be maximum at x = a if f(a) is greater than every other value of f(x) in the immediate neighborhood of x = a (i.e., f(x) ceases to increase but begins to increase at x = a. Similarly the minimum value of f(x) will be that value at x = b which is less than other values in the immediate neighborhood of x = b.<\/p>\n\n\n\n

 Maxima and minima of function of two variables<\/strong><\/h2>\n\n\n\n

As we know that the value of a function at maximum point is called maximum value of a function. Similarly the value of a function at minimum point is called minimum value of a function.<\/p>\n\n\n\n

The maxima and minima of a function is an extreme biggest and extreme smallest point of a function in a given range (interval) or entire region. Pierre de Fermat was the first mathematician to discover a general method for calculating maxima and minima of a function. The maxima and minima complement each other.<\/p>\n\n\n\n

Maxima and Minima of a function of one variables<\/strong><\/h2>\n\n\n\n

If  f(x) is a single valued function defined in a region R then <\/p>\n\n\n\n

Maxima is a maximum point  if and only if <\/p>\n\n\n\n

Minima is a minimum point  if and only if .<\/p>\n\n\n\n

Maxima and Minima of a function of two independent variables<\/strong><\/h2>\n\n\n\n

Let f(x, y) be a defined function of two independent variables.<\/p>\n\n\n\n

Then the point  x = a and y = b is said to be a maximum point of f(x, y) if<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

For all positive and negative values of h and k.<\/p>\n\n\n\n

Similarly the point x = a and y = b is said to be a minimum point of f(x, y) if<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

For all positive and negative values of h and k.<\/p>\n\n\n\n

Saddle point<\/strong><\/h2>\n\n\n\n

Critical points of a function of two variables are those points at which both partial derivatives of the function are zero. A critical point of a function of a single variable is either a local maximum, a local minimum, or neither. With functions of two variables there is a fourth possibility – a saddle point.<\/p>\n\n\n\n

Note<\/strong><\/h2>\n\n\n\n

1. Maximum and minimum values of a function occur alternatively<\/p>\n\n\n\n

2. Function may have several maximum and minimum values in an interval<\/p>\n\n\n\n

3. At some point the maximum value may be less than the minimum value<\/p>\n\n\n\n

4. The points at which a function has maximum or minimum value are called turning points and the maximum and minimum values are known as extreme values, or extremum or turning values.<\/p>\n\n\n\n

5. The values of x for which f(x) = 0 are often called critical values<\/p>\n\n\n\n

Criteria for maximum and minimum<\/strong><\/h2>\n\n\n\n

For a function y = f(x) to attain a maximum point at x = a,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

For minimum point-<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Conditions for max. and min-<\/strong><\/h2>\n\n\n\n

Necessary Condition<\/strong>– If a function f(x) is maximum or minimum at a point x = b and if f\u2019(b) exists then f\u2019 (b) = 0.<\/p>\n\n\n\n

Sufficient Condition-<\/strong> If b is a point in an interval where f(x) is defined and if f \u2018(b) = 0 and f\u2019\u2019(b) is not equal to 0, then  f(b) is maximum if f\u2019\u2019(b) <0 and is minimum if f\u2019\u2019(b) > 0. (The proof is not shown at present).<\/p>\n\n\n\n

Working Rule:<\/strong><\/p>\n\n\n\n

First derivative method<\/strong><\/h2>\n\n\n\n

To find the maximum or minimum point of a curve y = f(x).<\/p>\n\n\n\n

Find f \u2018 (x) and equate it to zero. From the equation f \u2018(x) = 0, find the value of x, say a and b.<\/p>\n\n\n\n

Here the number of roots of f \u2018(x) = 0 will be equal to the number of degree of f \u2018(x) = 0.<\/p>\n\n\n\n

Then find f \u2018(a \u2013 h) and f \u2018(a + h), then note the change of sign if any (here h is very small).<\/p>\n\n\n\n

If the change is from positive to negative, f(x) will be maximum at x = a. If again the change of sign is from negative to positive, f(x) will be maximum at x = a.<\/p>\n\n\n\n

Similarly for x = b.<\/p>\n\n\n\n

Second derivative method-<\/strong><\/h2>\n\n\n\n

First we find the first derivative of y = f(x) i.e dy\/dx and make it zero.<\/p>\n\n\n\n

From the equation  dy\/dx = 0 find the value of x say a and b.<\/p>\n\n\n\n

The again we find the second derivative of y or  <\/p>\n\n\n\n

Put x = a in , if at x = a is negative then the function is maximum at x = a and maximum value will be f(a).<\/p>\n\n\n\n

If the value of at x = a is positive, then the function is minimum and the minimum value will be f(a)<\/p>\n\n\n\n

Similarly we take for x = b.<\/p>\n\n\n\n

Example: Examine for maximum and minimum for the function <\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Sol.<\/p>\n\n\n\n

Here the first derivative is-<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

So that, we get-<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now we will get to know that the function is maximum or minimum at these values of x.<\/p>\n\n\n\n

For x = 3<\/p>\n\n\n\n

Let us assign to x, the values of 3 \u2013 h and 3 + h (here h is very small) and put these values at f(x).<\/p>\n\n\n\n

Then-<\/p>\n\n\n\n

  which is negative for h is very small<\/p>\n\n\n\n

which is positive.<\/p>\n\n\n\n

Thus f\u2019(x) changes sign from negative to positive as it passes through x = 3.<\/p>\n\n\n\n

So that f(x) is minimum at x = 3 and the minimum value is-<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

And f(x) is maximum at x = -3.<\/p>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/strong><\/p>\n\n\n\n