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Chapter 4 | Applications of Derivatives
Figure 4.38 Consider a twice-differentiable function f such that f ″ is continuous. Since f ′( a ) =0 and f ″( a ) <0, there is an interval I containing a such that for all x in I , f is increasing if x < a and f is decreasing if x > a . As a result, f has a local maximum at x = a . Since f ′( b ) =0 and f ″( b ) >0, there is an interval I containing b such that for all x in I , f is decreasing if x < b and f is increasing if x > b . As a result, f has a local minimum at x = b .
Theorem 4.11: Second Derivative Test Suppose f ′( c ) =0, f ″ is continuous over an interval containing c .
i. If f ″( c ) >0, then f has a local minimum at c . ii. If f ″( c ) <0, then f has a local maximum at c . iii. If f ″( c ) =0, then the test is inconclusive.
Note that for case iii. when f ″( c ) =0, then f may have a local maximum, local minimum, or neither at c . For example, the functions f ( x ) = x 3 , f ( x ) = x 4 , and f ( x ) =− x 4 all have critical points at x =0. In each case, the second derivative is zero at x =0. However, the function f ( x ) = x 4 has a local minimum at x =0 whereas the function f ( x ) =− x 4 has a local maximum at x , and the function f ( x ) = x 3 does not have a local extremum at x =0. Let’s now look at how to use the second derivative test to determine whether f has a local maximum or local minimum at a critical point c where f ′( c ) =0.
Example 4.20 Using the Second Derivative Test
Use the second derivative to find the location of all local extrema for f ( x ) = x 5 −5 x 3 .
Solution To apply the second derivative test, we first need to find critical points c where f ′( c ) =0. The derivative is
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