Some First Derivative FactsIf you have not already done so, you should review the tutorial on the First Derivative. Click here to see a picture that summarizes the First Derivative Test.
ConcavityThe Second Derivative Test provides a means of classifying relative extreme values by using the sign of the second derivative at the critical number. To appreciate this test, it is first necessary to understand the concept of concavity.
The graph of a function f
is strictly concave upward at the point
(c,f(c)) [or: f is convex at c] if
The graph of a function f
is strictly concave downward at the point
(c,f(c)) [or: f is concave at c] if
Concavity and the Second DerivativeThe important result that relates the concavity of the graph of a function to its derivatives is the following one:
Concavity Theorem:
If the function f is twice
differentiable at x = c, then the graph
of f is strictly concave upward at (c,f(c)) if La prova è analoga a quella del First Derivative Test, usando il polinomio di Taylor col resto nella forma di Lagrange invece che il teorema del valor medio (ossia il polinomio di Taylor, con resto di Lagrange, di ordine 1).
Example
Suppose f(x) = x^{3} 3x^{2} + x  2. Let's determine where the
graph of f is concave up and where it is concave down. Since f is
twicedifferentiable for all x, we use the result given above and first
determine that
Inflection PointsNotice in the example above, that the concavity of the graph of f changes sign at x = 1. Points on the graph of f where the concavity changes from uptodown or downtoup are called inflection points of the graph. The following result connects the concept of inflection point to the derivatives properties of the function:
Inflection Point Theorem:
If f′(c) exists and f″(c) changes sign at
x = c , then the point (c,f(c)) is an inflection point of
the graph of f. If
If we return to our example, where f(x) = x^{3}  3x^{2} + x 2, the Inflection Point Theorem verifies that the graph of f has an inflection
point at x = 1, since
The Second Derivative TestThe Second Derivative Test relates the concepts of critical points, extreme values, and concavity to give a very useful tool for determining whether a critical point on the graph of a function is a relative minimum or maximum.
The Second Derivative Test:
Suppose that c is a critical point
at which f′(c) = 0, that
Example:
Let's find and classify
the extreme values for the function
f with values f(x) = x^{3} 3x^{2} + x  2 that was introduced above. We find
that
NOTA Una funzione F si dice convessa (o con la concavità verso l'alto) in un dato intervallo I se ivi è definita e il suo epigrafico, ossia la figura formata dall'insieme dei punti con ascissa in I che stanno sopra al grafico di F, è convesso (ossia è tale che, presi comunque due punti in esso, anche il segmento che li congiunge sta tutto in esso). Se la funzione è derivabile due volte, questo equivale al fatto che F"(x) ≥ 0 per ogni x in I. F si dice concava (o con la concavità verso il basso) in I se F è convessa.
