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Use the given graph of a function π double prime to find the π₯-coordinates of the inflection points of π.
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So weβve been given the graph of the second derivative of a function and asked to use it to determine something about the function itself.
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First, weβll recall that, at an inflection point, the second derivative π double prime of π₯ is equal to zero.
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And now, this isnβt a sufficient condition for a point to be a point of inflection, as itβs also possible for the second derivative to be zero at a local minimum or a local maximum.
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But it does give us a starting place.
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From the given figure, we can see that π double prime of π₯ is equal to zero in three places, when π₯ is equal to one, when π₯ is equal to four, and when π₯ is equal to seven.
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So these are the π₯ coordinates of the three possible points of inflection of our function π.
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Now, letβs consider a little more about what we know about inflection points.
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There are points on the graph of a function where its concavity changes either from concave downward to concave upward or vice versa.
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We also recall that when a function is concave downward, its second derivative, π double prime of π₯, is negative.
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And when a function is concave oupward its second derivative is positive.
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At the inflection point itself, π double prime of π₯, is equal to zero, which is what weβve already used to determine our possible points of inflection.
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But the key point is that when a change in concavity occurs, there will also be a change in the sign of the second derivative.
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From the given figure, we can see that the sign of the second derivative changes from negative to positive around π₯ equals one and changes from positive to negative around π₯ equals seven.
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However, either side of π₯ equals four, the second derivative is positive, and so no change of sign occurs here.
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Hence, there is no change in the concavity of the function at π₯ equals four, but there is at π₯ equals one and π₯ equals seven.
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So we can conclude that our function π has inflection points at π₯ equals one and π₯ equals seven.