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Determine the inflection points of the curve π¦ equals π₯ squared plus two π₯ minus five.
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First of all, letβs remind ourselves what an inflection point is.
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Itβs a point where the concavity changes.
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We go from concave up to concave down or concave down to concave up.
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To identify inflection points, we use the second derivative.
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So, letβs visualize why the second derivative helps us to find inflection points.
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If this is a graph of a function π, not necessarily our function, we can see that this part of the graph is concave down and this part is concave up.
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So, the point in-between is an inflection point.
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Where the graph is concave down, the slope is decreasing.
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And where the graph is concave up, the slope is increasing.
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So, if we graph the derivative of this function π prime, which is the slope function, it would look like this.
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Decreasing, while the graph of π is concave down.
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And after the point of inflection, itβs increasing where the graph of π is concave up.
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What about the second derivative?
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Well, the slope of the graph of π prime of π₯ is negative before the point of inflection and itβs positive after.
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So, what does this tell us?
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A point of inflection occurs where the second derivative changes sign from negative to positive or positive to negative.
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So, thatβs why the second derivative helps us to find inflection points.
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Letβs go ahead and find our second derivative.
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To do this, letβs remember the general power rule that the derivative of ππ₯ to the power of π is πππ₯ to the power of π minus one.
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So, for our function of π₯, π₯ squared plus two π₯ minus five, π prime of π₯ equals two π₯ plus two.
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Thatβs because π₯ on its own is the same as π₯ to the power of one.
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So, two π₯ differentiates to two.
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And constants like negative five differentiate to zero.
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And now, we differentiate again to get our second derivative.
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This gives us π double prime of π₯ equals two.
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Remember, we said that, to find inflection points, weβre looking for points where the second derivative function changes sign.
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But we got a positive constant, so what does this mean?
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Well, π of π₯ is concave up where the second derivative is positive.
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As the second derivative is a constant, it canβt change sign.
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So, the graph of π¦ must always be concave up.
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So, as the concavity doesnβt change, we conclude that the curve has no inflection points.
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In fact, if we draw a sketch of this function, we can see that the curve is always concave up.
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There are no points of inflection on this curve.