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.**

## We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

- 30,000+ book summaries
- 20% study tools discount
- Ad-free content
- PDF downloads
- 300,000+ answers
- 5-star customer support

Already a member? Log in here.

Are you a teacher? Sign up now