Question

topic 5 lesson 1: practice
level 2: fill in the blanks/answer the questions

1. define and sketch an example of an inflection point.

Explanation:

An inflection point is a point on a curve where the concavity changes. That is, the second - derivative of the function changes sign at that point. For example, consider the function $y = x^{3}$. Its first - derivative $y'=3x^{2}$ and second - derivative $y'' = 6x$. The second - derivative is zero at $x = 0$. When $x<0$, $y''<0$ and the function is concave down. When $x>0$, $y''>0$ and the function is concave up. So, $(0,0)$ is an inflection point of $y = x^{3}$. To sketch it, draw the cubic function $y = x^{3}$ which passes through the origin $(0,0)$. The curve is concave down for negative $x$ values and concave up for positive $x$ values, changing concavity at the origin.

Answer:

An inflection point is a point on a curve where the concavity changes. For the function $y = x^{3}$, the point $(0,0)$ is an inflection point.