How to differeniate between local maxima & minima and global maxima & minima and extreme values?

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The extreme points of the graph of a function are those where the sign of the slope of the tangent changes. For a function f(x), at the extreme points f'(x) = 0. If at an extreme point for x = c the value of f(x) is less than the value of f(x) for values of x lying in an interval around c, there exists a minima at that extreme point. On the other hand if at an extreme point for x = c the value of f(x) is greater than the the value of f(x) for x lying in an interval around c, there is a maxima at f(x).

To differentiate a maxima from a minima look at the sign of the second derivative f''(x) at x = c. If f''(c) is negative the point is a maxima and if f''(c) is positive the point is a minima.

The global maxima and minima exist for some functions. At the global maxima the value of f(x) for x = c is greater than f(x) for any other value of x. Similarly for the global minima, the value of f(x) for x = c is less than the value of f(x) for any other value of x.

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