Understanding Stationary Points

00:00 4 marks

Explain the terms ‘critical point’ and ‘turning point’ as they relate to stationary points of a function. For each term, provide a mathematical definition and describe how it is identified. Furthermore, clarify the relationship between critical points and turning points.

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Hint

Think about the definition of a stationary point. How do critical points and turning points relate to the derivative of a function being zero or undefined? Consider examples of functions to illustrate your explanation.

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Complete Answer

A critical point of a function is a point in its domain where the first derivative is either zero or undefined. A turning point is a critical point where the function changes from increasing to decreasing (local maximum) or decreasing to increasing (local minimum). Therefore, all turning points are critical points, but not all critical points are turning points. For example, points of inflection can be critical points but are not turning points.

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