Step By Step Calculus » 13.5 - Critical Points, Concavity and Extrema

show_hiddenExpand All [+]
Synopsis
We are often interested in information about a graph along the lines of “what is the range of f(x)f(x)” (related to extrema, i.e. minima or maxima), and “how does f(x)f(x) behave around this point?” (related to concavity), a discussion of these two concepts is important.
In analyzing a graph, critical points are very important as they indicate possible local extrema. There are two kinds of critical points: those where f^\prime(x)f^\prime(x) is not defined, and those where f^\prime(x)=0f^\prime(x)=0. The latter are known as smooth critical points. Fermat’s theorem states that “ If ff has a local extremum at cc and f'(x)f'(x) exists at cc, then f'(c)=0.f'(c)=0.” One can state the following corollary using the Fermat’s theorem:
  • If ff has a local extremum at cc, then cc is a critical point or an endpoint of a closed interval in the domain.
However, not all critical points mark extrema. To find out the point of extrema, we can use the following test:
First Derivative Test: If cc is a critical point and f^{\prime }(x)f^{\prime }(x) exists near cc (but not necessarily at cc itself), then
\newcommand{\implies}{\Longrightarrow}\implies\newcommand{\implies}{\Longrightarrow}\implies A local maximum exists at cc if f^{\prime }(x)>0f^{\prime }(x)>0 for all x<cx<c ‘near’ cc and f^{\prime }(x)<0f^{\prime }(x)<0 for all x>cx>c ‘near’ cc.
\newcommand{\implies}{\Longrightarrow}\implies\newcommand{\implies}{\Longrightarrow}\implies A local minimum exists at cc if f^{\prime }(x)<0f^{\prime }(x)<0 for all x<cx<c ‘near’ cc and f^{\prime }(x)>0f^{\prime }(x)>0 for all x>cx>c ‘near’ cc.
\newcommand{\implies}{\Longrightarrow}\implies\newcommand{\implies}{\Longrightarrow}\implies otherwise, there can be either a local maxima or a local minima or neither at cc.
A complete procedure to find the extrema of a graph should be the following:
Step 1: (Differentiate) Find f'f'.
Step 2: (Important Points) Using ff and f'f', find
  • Singularities: The set SS of points at which the function ff is undefined on the given domain D_{f}D_{f}.
  • Critical points: The set C=\left\{ c:f^{\prime }(c)=0\text{ or }f^{\prime }(c)\text{ does not exist}\right\} C=\left\{ c:f^{\prime }(c)=0\text{ or }f^{\prime }(c)\text{ does not exist}\right\} . This set could contain intervals.
  • End points: The set EE comprising the endpoints of closed intervals in D_{f}D_{f}.
Step 3: (Regional Behaviour) Divide D_f\cap(S\cup C\cup E)^C\equiv D_f-(S\cup C\cup E)D_f\cap(S\cup C\cup E)^C\equiv D_f-(S\cup C\cup E) into regions and find behaviour of ff on each.
Step 4: (Local Extrema) Using the local behaviour of ff, identify points c\in Cc\in C and e\cup Ee\cup E as:
  • Local minima: if f^{\prime }(x)f^{\prime }(x) goes from \ominus \ominus to \oplus \oplus at ccorf(x)\geq f(e)f(x)\geq f(e) for all xx in an interval including ee.
  • Local maxima: if f^{\prime }(x)f^{\prime }(x) goes from \oplus \oplus to \ominus \ominus at ccorf(x)\leq f(e)f(x)\leq f(e) for all xx in an interval including ee.
  • Any or none of the above: if f^{\prime }(x)=0f^{\prime }(x)=0 or f^{\prime }(x)f^{\prime }(x) does not exist arbitrarily close to cc. In this case, try to graph the function around cc and go back to the basic definitions.
Step 5: (Global Extrema) Answer question asked noting that there might not be a global maximum or minimum. In particular, watch out for singularities.
The concept called ‘Concavity’ is another way to find out which critical points are extrema. A graph is concave up if it forms a ‘cup’ shape, and concave down if it forms a ‘cap’. Critical points in concave up regions are minima while critical points in concave down regions are maxima. Critical points are not necessarily extrema but also can be inflection points, where the graph changes concavity.

Concavity is controlled by the change in f^\primef^\prime, an increasing derivative (positive f^{(2)}f^{(2)}) indicates concave up, and a decreasing derivative (negative f^{(2)}f^{(2)}) indicates concave down.
Second Derivative Test: If cc is a critical point and f^{(2)}(x)f^{(2)}(x) exists at and nearcc and is continuous at cc, then
\newcommand{\implies}{\Longrightarrow}\implies\newcommand{\implies}{\Longrightarrow}\implies A local maximum exists at cc if f^{(2)}(c)<0f^{(2)}(c)<0. Under the conditions the given cc would be a smooth critical point in a concave down region.
\newcommand{\implies}{\Longrightarrow}\implies\newcommand{\implies}{\Longrightarrow}\implies A local minimum exists at cc if f^{(2)}(c)>0f^{(2)}(c)>0. Under the conditions the given cc would be a smooth critical point in a concave up region.
Second derivative test can not conclude whether a local extrema exists at a critical point cc if f^{(2)}(x)f^{(2)}(x) does not exist at cc or f^{(2)}(c)=0f^{(2)}(c)=0. The points cc where f^{(2)}(x)f^{(2)}(x) does not exist or is equal to 00 are the possible inflection points and we denote this set of points by II. A point i\in Ii\in I is an inflection point if f^{(2)}f^{(2)} changes its sign (equivalently f'f' becomes increasing to decreasing or vice versa) as it moves from one side of ii to the other side of ii.