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Concavity and SecondDerivative Test with firstderivative
test
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Concavity
Concave UP
Concave DOWN
Inflection point:Where concavitychanges
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Inflection Point
Consider the slope as curve changesthrough concave up to concave
down
Slope starts
negative
Becomesless negative
Slope becomes(horizontal) zeroSlope becomespositive, then
more positive
At inflection point
slope reachesmaximum
positive value After inflection
point, slopebecomes less
positive
Graph of the slope
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Inflection Point
What could you say about the slope functionwhen the original
function has an inflectionpoint
Graph of the slope
Slope function has a maximum (or minimum
Thus second derivative = 0
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Second Derivative
This is really the rate of change of the slope When the original
function has a relative
minimumSlope is increasing (left to right) and goesthrough
zeroSecond derivative is positiveOriginal function is concave
up
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Second Derivative
If the second derivative f (x) = 0 The slope is neither
increasing nor decreasing
If f (x) = 0 at the same place f (x) = 0 The 2 nd derivative
test failsYou cannot tell what the function is doing
4( ) f x x2''( ) 12 f x xNot an inflection
point
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SummaryInterval f(x) f(x) f(x) Information obtained
x
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Try this!
f(x)=x 6/6 -5x 4/4 +2x 2
f (x) = ?
f (x) = ?
Where are relative max, min, inflection point?
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