WEBVTT
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So we're told in part a that the derivative of
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our function F. Of X is less than zero
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. And also the second derivative of our function F
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. Of X is less than zero for all X
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. And what we want to do is use this
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information to try and sketch a potential graph of this
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function. And so what this tells us since our
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first derivative is less than zero for all X.
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That means that we're decreasing for the entirety of our
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function. So we're decreasing the entirety of our function
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. And since we know that the second derivative effects
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is less than zero, we also know that we
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are concave down for the entirety of our functions.
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The second derivative tells us about con cavity in the
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first derivative tells us about the slope or whether we're
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increasing or decreasing. And so what we want to
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do is we want to draw a graph that is
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both concave down and decreasing for all of its domain
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. So how we do this is I'm just going
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to start here and make sure and make sure that
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we're concave down and we're decreasing this entirety of this
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curve and then just put an air over there and
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an arrow like that where we're never actually going to
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get a horizontal um assume toad or slope or anything
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like that. We're just going to be concave down
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this entire way, but we're also decreasing. So
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this is what a sketch of the graph given these
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conditions would look like. And for part B we're
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told that F prime of X is greater than zero
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and F double prime of X is also greater than
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zero for all of the X. So in this
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case we're going to be concave up and increasing for
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the entirety of our function. So we want to
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draw a graph that is both concave up. Actually
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. Drawed on this type of graph that is concave
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up and increasing for the entirety of the function.
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So I'm just going to kind of start like like
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this and go like that and make sure that we're
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concave up so were curved upwards like a U.
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Shaped A upwards you shape. And we're also increasing
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for the entirety of our function. So a graph
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like this could be a potential sketch for a function
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given these conditions.