Questions: (c) Find the open interval(s) on which f is concave upward. (Enter your answer using interval notation.) (d) Find the interval(s) on which f is concave downward. (Enter your answer using interval notation.) (e) Find the coordinates of the point(s) of inflection. (x, y)=( x)

Solution Steps

To determine the concavity of the function \( f \) and find the points of inflection, we need to follow these steps:

1. Find the second derivative \( f''(x) \) of the function \( f(x) \).
2. Determine the critical points by setting \( f''(x) = 0 \) and solving for \( x \).
3. Analyze the sign of \( f''(x) \) on the intervals determined by the critical points to identify where \( f \) is concave upward (where \( f''(x) > 0 \)) and concave downward (where \( f''(x) < 0 \)).
4. Identify the points of inflection as the points where \( f''(x) \) changes sign.

To find the intervals of concavity and points of inflection, we need to determine the second derivative of the function \( f(x) \).

Set the second derivative equal to zero and solve for \( x \) to find the critical points.

Use the critical points to divide the number line into intervals. Test the sign of the second derivative in each interval to determine concavity.

• If \( f''(x) > 0 \) on an interval, \( f \) is concave upward on that interval.
• If \( f''(x) < 0 \) on an interval, \( f \) is concave downward on that interval.

Points of inflection occur where the concavity changes, i.e., where \( f''(x) \) changes sign.

Final Answer