Question

to determine the intervals where a function is concave up and concave down, the first step is to find all the x values where (select all that are needed):

• $f(x)=0$
• $f(x)$ is undefined
• $f(x)=0$
• $f(x)$ is undefined
• $f(x)=0$
• $f(x)$ is undefined

the second step is to use these values to form intervals, and on each interval:

• test the sign of $f(x)$ in the interval to see if it is positive or negative
• test the sign of $f(x)$ in the interval to see if it is positive or negative
• test the sign of $f(x)$ in the interval to see if it is positive or negative

we do this because:

• when $f(x)$ is positive it tells us $f(x)$ is concave up on that interval, and when $f(x)$ is negative it tells us $f(x)$ is concave down on that interval
• when $f(x)$ is positive it tells us $f(x)$ is concave up on that interval, and when $f(x)$ is negative it tells us $f(x)$ is concave down on that interval
• when $f(x)$ is positive it tells us $f(x)$ is concave up on that interval, and when $f(x)$ is negative it tells us $f(x)$ is concave down on that interval

Explanation:

Step1: Recall concavity rules

To find intervals of concavity, we first need to find where $f''(x)=0$ or $f''(x)$ is undefined. These $x -$ values are called inflection - point candidates. So the needed $x$ values are when $f''(x)=0$ and $f''(x)$ is undefined.

Step2: Recall interval - testing for concavity

After getting the values from Step 1, we form intervals and test the sign of $f''(x)$ in each interval. If $f''(x)>0$ on an interval, the function $f(x)$ is concave up on that interval. If $f''(x)<0$ on an interval, the function $f(x)$ is concave down on that interval.

Answer:

First part: $f''(x)$ is undefined, $f''(x)=0$
Second part: Test the sign of $f''(x)$ in the interval to see if it is positive or negative
Third part: When $f''(x)$ is positive it tells us $f(x)$ is concave up on that interval, and when $f''(x)$ is negative it tells us $f(x)$ is concave down on that interval