(b) Determine the point of inflection of the curve \( y=x^{1 / 3} \).

To find the point of inflection of the curve \( y = x^{1/3} \), we first need to calculate the second derivative. The first derivative is \( y' = \frac{1}{3} x^{-2/3} = \frac{1}{3\sqrt[3]{x^2}} \). The second derivative is \( y'' = -\frac{2}{9} x^{-5/3} \). A point of inflection occurs where the second derivative changes sign, which typically means finding points where it equals zero or is undefined. Here, \( y'' = 0 \) does not apply as it never equals zero, but it is undefined at \( x=0 \). To ensure \( x=0 \) is indeed an inflection point, we check the sign of \( y'' \) around \( 0 \). For \( x < 0 \), \( y'' > 0 \), and for \( x > 0 \), \( y'' < 0 \). Hence, the point of inflection is at \( (0,0) \). Now you have the mathematical result; let’s dive into some fun background and application of this concept! The study of inflection points goes back to the 17th century with the work of mathematicians like Isaac Barrow and Gottfried Wilhelm Leibniz, who laid the groundwork for calculus. Inflection points are critical in fields such as physics and engineering, where they can indicate changes in motion or the structure of a physical system. Identifying these points helps predict behavior in dynamic systems! In real-life scenarios, you may often encounter curves that model everything from profit margins in business to trajectories in sports. Understanding inflection points can help you optimize these models. For instance, if a company’s profit curve has an inflection point, it signifies a shift in growth strategy—like moving from a startup phase to a more mature phase. Spotting that can significantly impact decision-making!