How do you use derivatives to categorise stationary points?

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When investigating graphs, you will often be asked to pick out features of the graph; stationary points being the most popular. You will need to know that a stationary point on f(x) can be found by solving the following equation: f'(x)=0.

Once you have found the stationary points, you will need to find the second derivative of the graph, also known as f''(x). By finding the values of f''(x) at the x-coordinates where stationary points exist, you can categorise the stationary points.

If f''(x) > 0, then the stationary point is a minimum point.

If f''(x) < 0, then the stationary point is a maximum point.

If f''(x) = 0, then the stationary point is a point of inflection.

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