Explain the whole concept of Maxima and Minima!!!!!

Maxima and Minima, known collectively as extrema, are the largest value (maximum) or smallest value (minimum), that a function takes in a point either within a given neighbourhood (local extremum) or on the function domain in its entirety.

Check out this Figure :

 Here point A is Local maxima and point B is Local minima .At each of these points ,the tangent of the curve is parallel to the x-axis .So the derivative of the function is zero.Both of these points are stationary points of the function.The term local is used thus these points are the maximum and minimum in this particular region .There may be others outside this region.

Further on the gradients to the curve are :

#  to the left of A ,the gradeints are positive (+)

#  between A and B the gradeints are negative (-)

#  to the right of B , the gradients are positive (+)

About the local maximum point A, the gradient changes from positive to Zero to negative . The gradeint is therefore decreasing.

About the local minimum point B, the gradient changes from negative  to Zero to positive . The gradeint is therefore increasing .