A tutorial on how to use calculus theorems using first and second derivatives to determine whether a function has a relative maximum or minimum or neither at a given point. These are some of the most important theorems in problem solving.

Theorem 1 - Stationary Point

If function \(f\) has a relative minimum or maximum at \(x = a\), we either have \(f'(a) = 0\) (stationary point) or \(f'(a)\) does not exist.
As an example, the graph of \(f\) and its derivative \(f'\) are shown above. At the maximum (\(x = 2\)) and the minimum (\(x = -2\)) of \(f\), \(f' = 0\).

Theorem 2 - First Derivative Test

Let \(f\) be a continuous function.
2.a - If \(f'(a) = 0\) or \(f'(x)\) does not exist at \(x = a\) and if \(f'(x) \lt 0\) to the left of \(a\) and \(f'(a) > 0\) to the right of \(a\), then \(f\) has a relative minimum at \(x = a\).
2.b - If \(f'(a) = 0\) or \(f'(x)\) does not exist at \(x = a\) and if \(f'(x) > 0\) to the left of \(a\) and \(f'(a) \lt 0\) to the right of \(a\), then \(f\) has a relative maximum at \(x = a\).
Both 2.a and 2.b can clearly be verified using the graph in figure 1 above.
3.b - If \(f'(a)\) has the same sign to the left and to the right of \(x = a\), then \(f\) does not have a minimum or maximum.
The graph in figure 2 below shows the graph of a function \(f\) that has neither a minimum nor maximum and its first derivative does not change sign.

Theorem 3 - First and Second Derivatives Test

Suppose that both \(f'\) and \(f''\) exists at \(x = a\) and that \(f'(a) = 0\) (stationary point).
3.a - If \(f''(a) > 0\), \(f\) has a relative minimum at \(x = a\).
3.b - If \(f''(a) \lt 0\), \(f\) has a relative maximum at \(x = a\).
3.c - If \(f''(a) = 0\), no conclusion can be made, use theorem 2 above.
Below are shown the graph of function \(f\), its first derivative \(f'\), and its second derivative \(f''\). We can easily check that:
1) at the point (\(x = -2\)) where \(f'(-2) = 0\) and \(f''(-2) \lt 0\), \(f\) has a maximum
2) also at the point (\(x = 2\)) where \(f'(2) = 0\) and \(f''(2) > 0\), \(f\) has a minimum at \(x = 2\).

More References and Links to Calculus Problems

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Calculus, 11th Edition, ISBN: 978-1-118-88613-7, Jun 2016 , Howard Anton, Irl C. Bivens, Stephen Davis ,