# 4.4 | Concavity and the Second Derivative Test

Definition of Concavity
Let $$f$$ be differentiable on an open interval $$I$$. The graph of $$f$$ is concave upward on $$I$$ when $$f’$$ is increasing on the interval and concave downard on $$I$$ when $$f’$$ is decreasing on $$I$$
If $$(c,f(c))$$ is a point of inflection of the graph of $$f$$, then either $$f^{\prime \prime}(c) = 0$$ or $$f^{\prime \prime}$$ does not exist at $$c$$.
Test for Concavity
Let $$f$$ be a function whose second derivative exists on an open interval $$I$$.

1. If $$f^{\prime \prime} (x) >0$$ for all $$x$$ in $$(a,b)$$, then $$f$$ is concave upward on $$I$$.
2. If $$f^{\prime \prime} (x) < 0$$ for all $$x$$ in $$(a,b)$$, then $$f$$ is concave downward on $$I$$.
Definition of Point of Inflection
Let $$f$$ be a function that is continuous on an open interval, and let $$c$$ be a point in the interval. If the graph of $$f$$ has a tangent line at this point $$(c,f(c))$$ and the graph of $$f$$ changes concavity at $$c$$, then the point $$(c,f(c))$$ is a point of inflection.
The Second Derivative Test
Let $$f$$ be a function such that $$f'(c) = 0$$ and the second derivative of $$f$$ exists on an open interval containing $$c$$.

1. If $$f^{\prime \prime} (c) >0$$, then $$f$$ has a relative minimum at $$(c,f(c))$$.
2. If $$f^{\prime \prime} (c) < 0$$, then $$f$$ has a relative maximum at $$(c,f(c))$$.

If $$f^{\prime \prime}(c) = 0$$, then the test is inconclusive. That is, $$f$$ may or may not have a relative extrema. In such cases, you can use the First Derivative Test.