CONCAVIT

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DERIVAT

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WARM UP

Given derivative graph below, find a. intervals where the original function is increasing b. intervals where the original function is decreasing c. x-coordinates of the local maximums and minimums

of the function

WHAT IS CONCAVITY?

Concavity is another physical interpretation of a function

SECOND DERIVATIVE = CONCAVITY

The second derivative tells us intervals where the function is concave up and concave down.

INFLECTION POINTS

Inflection points are the points where the graph changes concavity

INTERVALS OF CONCAVE UP AND DOWN

Find the intervals of concavity for the function

Step 1: Find the second derivative, f”(x) = 0

Step 2: Determine the x-coordinates of the points of inflection

Step 3: Test the concavity for each interval in f”(x)

AP PROBLEM

SECOND DERIVATIVE TEST FOR EXTREMA

TRY IT – SECOND DERIVATIVE TEST

Find the relative extrema of the function.

Step 1: Find the critical numbers of the function, where f’(x) = 0

Step 2: Find the second derivative and test each x-value to see the concavity at each point.

Step 3: Plug x-values into original to find coordinates

COMPARISON OF 1ST AND 2ND DERIVATIVE

=0 Intervals Extrema Test

1st Derivative Critical Points(m=0)

Increasing/Decreasing

Use critical points and intervals of increasing/decreasing

2nd Derivative Inflection Points(concavity changes)

Concave Up/Down

Use critical points and concavity