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.
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