Can a function have only a points of maxima but no points of minima?
For a function f(x), the extreme points are located at points where f'(x) = 0. If the value of f''(x) at these points is negative they are points of maxima and if the value of f''(x) at these points is positive they are points of minima.
For a function f(x) to have only a point of maxima but no point of minima, the value of f''(c) where c is the solution of f'(x) = 0 should be negative. And at no extreme point should the value of the second derivative be positive.
An example of this would be f(x) = x - x^2
f'(x) = 1 - 2x
1 - 2x = 0
=> x = 1/2
f''(x) = -2
The function has a maxima at (1/2, 1/4) but no minimum points. This is to do with the fact that if you see the graph of the function there are no points where the value of the function is greater that it is at x = 1/2. But it has no point where the value of f(x) is lower than it is at any other point.
It is possible for a function to have a point of maxima and no points of minima.