5.6 Minimum and maximum values  (Page 2/6)

Critical points

Critical points are those points where minimum or maximum of a function can occur. We see that minimum and maximum of function can occur at following points :

(a) Points on the graph of function, where derivative of function is zero. At these points, function is continuous, limit of function exists and tangent to the curve is parallel to x-axis.

(b) Points where function is continuous but not differentiable. Limit of function exits at those points and are equal to function values. Consider, for example, the corner of modulus function graph at x=0. Minimum of function exist at the corner point i.e at x=0.

(c) Points where function is discontinuous (note that discontinuous is not undefined). A function has function value at the point where it is discontinuous. Neither limit nor derivative exists at discontinuities. Example : piece-wise defined functions like greatest integer function, fraction part function etc.

We can summarize that critical points are those points where (i) derivative of function does not exist or (ii) derivative of function is equal to zero. The first statement covers the cases described at (b) and (c) above. The second statement covers the case described at (a). We should, however, be careful to interpret definition of critical points. These are points where minimum and maximum “can” exist – not that they will exist. Consider the graph shown below, which has an inflexion point at “A”. The tangent crosses through the graph at inflexion point. In the illustration, tangent is also parallel to x-axis. The derivative of function, therefore, is zero. But “A” is neither a minimum nor a maximum.

Thus, minimum or maximum of function occur necessarily at critical points, but not all critical points correspond to minimum or maximum.

Note : We need to underline that concept of critical points as explained above is different to the concept of critical points used in drawing sign scheme/ diagram.

Graphical view

There are mathematical frameworks to describe and understand nature of function with respect to minimum and maximum. We can, however, consider a graphical but effective description that may help us understand occurrence of minimum and maximum values. We need to understand one simple fact that we can have graphs of any nature except for two situations :

1: function is not defined at certain points or in sub-intervals.

2: function can not be one-many relation. In this case, the given relation is not a function in the first place.

Clearly, there exists possibility of minimum and maximum at all points on the continuous portion of function where derivative is zero and at points where curve is discontinuous. This gives us a pictorially way to visualize where minimum and maximum can occur. The figure, here, shows one such maximum value at dicontinuity.

Relative or local minimum and maximum

The idea of local or relative minimum and maximum is clearly understood from graphical representation. The minimum function value at a point is least in the immediate neighborhood where minimum occurs. A function has a relative minimum at a point x=m, if function values in the immediate neighborhood on either side of point are less than the value at the point. To be precise, the immediate neighborhood needs to be infinitesimally close. Mathematically,