Find all relative extrema and points of inflection for the function
. Be sure to give coordinate pairs for each point. You do not need to draw the graph.
Foundations:
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Since our function is a polynomial, the relative extrema occur when the first derivative is zero. We then have two choices for finding if it is a local maximum or minimum:
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Second Derivative Test: If the first derivative at a point is , and the second derivative is negative (indicating it is concave-down, like an upside-down parabola), then the point is a local maximum.
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On the other hand, if the second derivative is positive, the point is a local minimum. You can also use the first derivative test, but it is usually a bit more work! For inflection points, we need to find when the second derivative is zero, as well as check that the second derivative "splits" on both sides.
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Solution:
Step 1:
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Find the first and second derivatives: Based on our function, we have
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Similarly, from the first derivative we find
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Step 2:
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Find the roots of the derivatives: We can rewrite the first derivative as
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from which it should be clear we have roots and .
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On the other hand, for the second derivative, we have
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This has a single root: .
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Step 3:
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Test the potential extrema: We know that are the candidates. We check the second derivative, finding
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while
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Note that
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while
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By the second derivative test, the point is a relative minimum, while the point is a relative maximum.
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Step 4:
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Test the potential inflection point: We know that . On the other hand, it should be clear that if , then . Similarly, if , then . Thus, the second derivative "splits" around (i.e., changes sign), so the point is an inflection point.
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Since
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our inflection point is
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Final Answer:
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There is a local minimum at , a local maximum at and an inflection point at
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