Maxima and Minima

To find maximum and minimum value of a given function.

Differentiate the function first and then find the values of x at which the function becomes zero these are the possible points of Maximum and minimum values of the function then apply any of the following test

(1) 1st derivative test: If at any of the above values of x dy/dX changes its sign from +ve to -ve then that is a point of local Maxima.

And if dy/dX changes sign from -ve to +ve that is a point of local minima.

(2) 2nd derivative test: Find the 2nd derivative and put the above values of x obtained if it comes -ve that is a point of local Maxima and if it comes +ve that's a point of local minima.

Note: there can be more than 1 point of local Maxima and Minima

Global Maximum and Minimum: put all the values of x obtained in the function and check which one is largest, that value correspond to Global Maxima and that vale is global maximum. Similarly check for Global minima.

Note:sometimes 2nd derivative test fails i.e it becomes zero that that point is called point of inflection. Then check with first derivative test.