JEE Main & Advanced Mathematics Applications of Derivatives Greatest and Least Values of a Function Defined on an Interval \[[a,\,\,b]\]

Greatest and Least Values of a Function Defined on an Interval \[[a,\,\,b]\]

Category : JEE Main & Advanced

By maximum (or minimum) or local maximum (or local minimum) value of a function \[f(x)\] at a point \[c\in [a,b]\] we mean the greatest (or the least) value in the immediate neighbourhood of \[x=c\]. It does not mean the greatest or absolute maximum (or the least or absolute minimum) of \[f(x)\]in the interval \[[a,\,b]\].

 

 

A function may have a number of local maxima or local minima in a given interval and even a local minimum may be greater than a relative maximum.

 

 

Thus a local maximum value may not be the greatest (absolute maximum) and a local minimum value may not be the least (absolute minimum) value of the function in any given interval.

 

 

However, if a function \[f(x)\] is continuous on a closed interval \[[a,\,b]\], then it attains the absolute maximum (absolute minimum) at critical points, or at the end points of the interval \[[a,\,b]\]. Thus, to find the absolute maximum (absolute minimum) value of the function, we choose the largest and smallest amongst the numbers \[f(a),f({{c}_{1}}),f({{c}_{2}}),....,f({{c}_{n}}),f(b)\], where \[x={{c}_{1}},{{c}_{2}},....,{{c}_{n}}\] are the critical points.



LIMITED OFFER HURRY UP! OFFER AVAILABLE ON ALL MATERIAL TILL TODAY ONLY!

You need to login to perform this action.
You will be redirected in 3 sec spinner