Chapter 4 | Applications of Derivatives
401
f ′( x ) =5 x 4 −15 x 2 . Therefore, f ′( x ) =5 x 4 −15 x 2 =5 x 2 ⎛ ⎞ ⎠ =0 when x =0, ± 3. To determine whether f has a local extrema at any of these points, we need to evaluate the sign of f ″ at these points. The second derivative is f ″( x ) =20 x 3 −30 x =10 x ⎛ ⎝ 2 x 2 −3 ⎞ ⎠ . In the following table, we evaluate the second derivative at each of the critical points and use the second derivative test to determine whether f has a local maximum or local minimum at any of these points. ⎝ x 2 −3
f ″ ( x )
Conclusion
x
Local maximum
− 3 −30 3
0
0
Second derivative test is inconclusive
Local minimum
3
30 3
By the second derivative test, we conclude that f has a local maximum at x =− 3 and f has a local minimum at x = 3. The second derivative test is inconclusive at x =0. To determine whether f has a local extrema at x =0, we apply the first derivative test. To evaluate the sign of f ′( x ) =5 x 2 ⎛ ⎝ x 2 −3 ⎞ ⎠ for x ∈ ⎛ ⎝ − 3, 0 ⎞ ⎠ and x ∈ ⎛ ⎝ 0, 3 ⎞ ⎠ , let x =−1 and x =1 be the two test points. Since f ′(−1) <0 and f ′(1) <0, we conclude that f is decreasing on both intervals and, therefore, f does not have a local extrema at x =0 as shown in the following graph.
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