IXL - Identify the graph of the second derivative from the graph of the function (Calculus practice)

This is the graph of a function

Which of the following is the graph of

′′

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question

This is the graph of a function

Which of the following is the graph of

′′

key idea

A graph is concave up on an interval if the graph lies above all of its tangent lines on that interval.

A graph is concave down on an interval if the graph lies below all of its tangent lines on that interval

The concavity of a graph tells you about the behavior of the second derivative.

If the graph of
f
is concave up on an interval, then
f
′′
(
x
)
≥
0
on that interval.
If the graph of
f
is concave down on an interval, then
f
′′
(
x
)
≤
0
on that interval.

solution

To find the graph of

′′

look at the graph of

Identify where the graph of

is concave up and determine what this means about the behavior of

′′

The graph of

is concave up on the interval

(

–

)

So,

′′

(

)

≥

on this interval.

Next, identify where the graph of

is concave down and determine what this means about the behavior of

′′

(

)

The graph of

is concave down on the intervals

(

–

∞

–

)

and

(

∞

)

So,

′′

(

)

≤

on these intervals.

You now know where

′′

(

)

≥

and where

′′

(

)

≤

So, the graph of

′′

must lie in the following shaded regions.

This graph is the only answer choice that lies only within the shaded regions.

So, this is the graph of

′′

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