point at which it is not even differentiable (Example ). Fig . Example Find the maximum and the minimum values, if any, of the function f given by f ( x ) = x , x ∈ R . Solution From the graph of the given function (Fig .
), we have f ( x ) = if x = . Also f ( x ) ≥ , for all x ∈ R . Therefore, the minimum value of f is and the point of minimum value of f is x = . Further, it may be observed from the graph of the function that f has no maximum value and hence no point of maximum value of f in R .
A Note If we restrict the domain of f to [– , ] only, then f will have maximum value(– ) = at x = – . Example Find the maximum and minimum values of f , if any, of the function given by f ( x ) = | x |, x ∈ R . Solution From the graph of the given function (Fig . ) , note that f ( x ) ≥ , for all x ∈ R and f ( x ) = if x = .
Therefore, the function f has a minimum value and the point of minimum value of f is x = . Also, the graph clearly shows that f has no maximum value in R and hence no point of maximum value in R .