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Use the given graph of π to find all possible intervals on which π is concave downward.
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And then we have the graph of π plotted within the first quadrant.
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So letβs begin by reminding ourselves what it means for a function π to be concave downward.
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Letβs begin by thinking about the algebraic definition.
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Given a graph of π, it will be concave downward if π prime is decreasing, in other words, if the rate of change of the slope of the function is decreasing.
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But what does this mean for the graph of the function?
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How can we quickly identify the intervals on which the graph is conclave downward?
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To think about this graphically, we need to think about what the tangent lines to the curve look like.
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If π prime is decreasing, the graph of π is concave downwards and the tangent lines will all lie above the curve of π at a given point.
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For instance, letβs take the very first portion of our graph.
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We can draw the tangents at approximately π₯ equals 0.5, π₯ equals one, and π₯ equals 1.5.
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We see that the tangents all lie above the curve of our function π.
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In particular, we notice that the slope of each line must be decreasing.
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So this first part is certainly concave downwards.
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But where does it stop?
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Well, we notice that if we attempt to draw the tangent lines to the curve at approximately π₯ equals 2.4, π₯ equals three, and π₯ equals 3.3, the tangent lines lie below the curve of the function.
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So at this point, π is actually concave upward.
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We might observe that the change over point here is at π₯ equals two.
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So the curve is concave downwards between π₯ equals zero and π₯ equals two.
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But is it concave downwards at any other locations?
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Well, yes.
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The slope of π is actually decreasing over a second interval.
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Specifically, this appears to happen at π₯ equals four, and it stops at π₯ equals five, where the slope is in fact zero.
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Since π prime is neither decreasing nor increasing at points where itβs equal to zero, then the second interval is from four to five.
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And so weβve identified the intervals on which π is concave downward.
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Now to save us time in future, we can observe that when π is concave downwards, we have something that looks a little bit like a cave.
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When itβs concave upward, we have something that looks like a cup.
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So π is concave downward on the open interval from zero to two and the open interval from four to five.