First and Second Derivatives Theorems

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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