What is required to determine the point of inflection of a function?

What is required to determine the point of inflection of a function?

• A. Find the first derivative and set it to zero
• B. Calculate the limits of the function
• C. Find the second derivative and solve for x
• D. Identify the roots of the function

Answer: (C)

To determine the point of inflection of a function, one needs to find the second derivative of that function and solve for the variable. A point of inflection occurs where the curvature of the graph changes, which is identified by changes in the sign of the second derivative. When the second derivative is zero or undefined at some point \( x = c \), it indicates that there may be a change in concavity at that point. However, to confirm that it is indeed a point of inflection, one should also check whether the second derivative changes signs before and after \( x = c \). Therefore, finding the second derivative is essential in determining both potential points of inflection and confirming their nature. In contrast, setting the first derivative to zero typically identifies critical points, which could indicate local maxima or minima but do not directly address changes in concavity. Calculating limits of the function is important for understanding its behavior at certain values but does not directly pertain to identifying points of inflection. Identifying the roots of the function focuses on where the function intersects the x-axis, which is not relevant for determining points of inflection.

To determine the point of inflection of a function, one needs to find the second derivative of that function and solve for the variable. A point of inflection occurs where the curvature of the graph changes, which is identified by changes in the sign of the second derivative.

When the second derivative is zero or undefined at some point ( x = c ), it indicates that there may be a change in concavity at that point. However, to confirm that it is indeed a point of inflection, one should also check whether the second derivative changes signs before and after ( x = c ). Therefore, finding the second derivative is essential in determining both potential points of inflection and confirming their nature.

In contrast, setting the first derivative to zero typically identifies critical points, which could indicate local maxima or minima but do not directly address changes in concavity. Calculating limits of the function is important for understanding its behavior at certain values but does not directly pertain to identifying points of inflection. Identifying the roots of the function focuses on where the function intersects the x-axis, which is not relevant for determining points of inflection.