# 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.

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