Second derivative test: Difference between revisions

Revision as of 15:26, 2 May 2012

Statement

What this test is for

This test is a partial test (i.e., it may be inconclusive) for determining whether a given critical point for a function is a point of local minimum, point of local maximum, or neither.

What the test says

Suppose $f$ is a function and $c$ is a point in the interior of the domain of $f$ , i.e., $f$ is defined on some open interval containing $c$ . Suppose, further, that $f''$ , i.e., the second derivative of $f$ , exists at $c$ . Suppose also that $f'(c)=0$ , so $c$ is a critical point for $f$ . Then:

Hypothesis	Conclusion
$f''(c)<0$	$f$ attains a local maximum value at $c$ (the value is $f(c)$ )
$f''(c)>0$	$f$ attains a local minimum value at $c$ (the value is $f(c)$ )
$f''(c)=0$	The test is inconclusive. $f$ may attain a local maximum value, a local minimum value, have a point of inflection, or have some different behavior at the point $c$ .

Relation with critical points

The second derivative test is specifically used only to determine whether a critical point where the derivative is zero is a point of local maximum or local minimum. Note in particular that:

For the other type of critical point, namely that where $f'$ is undefined, the second derivative test cannot be used.
Since point of local extremum implies critical point, we don't have to worry about points that are not critical points -- none of them will give local extrema.

Related tests

Strength of the test

Second derivative test requires twice differentiability at but not around the point

The second derivative test can be applied at a critical point $c$ for a function $f$ only if $f$ is twice differentiable at $c$ . This in particular forces $f$ to be once differentiable around $c$ .

However, the test does not require $f''$ to be defined around $c$ or to be continuous at $c$ .

@@ Line 5: / Line 5: @@
 This test is a partial test (i.e., it may be inconclusive) for determining whether a given [[critical point]] for a [[function]] is a point of local minimum, point of local maximum, or neither.
-===What the test states===
+===What the test says===
 Suppose <math>f</math> is a [[function]] and <math>c</math> is a point in the interior of the domain of <math>f</math>, i.e., <math>f</math> is defined on some [[open interval]] containing <math>c</math>. Suppose, further, that <math>f''</math>, i.e., the [[second derivative]] of <math>f</math>, exists at <math>c</math>. Suppose also that <math>f'(c)=0</math>, so <math>c</math> is a [[critical point]] for <math>f</math>. Then:
@@ Line 18: / Line 18: @@
 | <math>f''(c) = 0</math> || The test is inconclusive. <math>f</math> may attain a local maximum value, a local minimum value, have a [[point of inflection]], or have some different behavior at the point <math>c</math>.
 |}
+===Relation with critical points===
+The second derivative test is specifically used ''only'' to determine whether a [[critical point]] where the derivative is zero is a point of local maximum or local minimum. Note in particular that:
+* For the other type of critical point, namely that where <math>f'</math> is undefined, the second derivative test cannot be used.
+* Since [[point of local extremum implies critical point]], we don't have to worry about points that are not critical points -- none of them will give local extrema.
 ==Related tests==
@@ Line 23: / Line 30: @@
 * [[First derivative test]]
 * [[Higher derivative test]]s
+==Strength of the test==
+===Second derivative test requires twice differentiability at but not around the point===
+The second derivative test can be applied at a critical point <math>c</math> for a function <math>f</math> only if <math>f</math> is ''twice'' differentiable at <math>c</math>. This in particular forces <math>f</math> to be once differentiable ''around'' <math>c</math>.
+However, the test does ''not'' require <math>f''</math> to be defined around <math>c</math> or to be continuous at <math>c</math>.