Problem 54 - Essential Calculus

Question

(a) Graph \(f(x)=\sqrt[3]{x}\) and identify the inflection point. (b) Does \(f^{\prime \prime}(x)\) exist at the inflection point? Explain.

Step-by-Step Solution

Answer

The function \(f(x) = \sqrt[3]{x}\) has an inflection point at \(x = 0\). However, the second derivative \(f^{\prime\prime}(x)\) does not exist at this point.

Step 1: Derive the function

The derivative of the function \(f(x) = \sqrt[3]{x}\) is obtained by using the power rule for differentiation. The power rule states that the derivative of \(x^n\), where n is any real number, is \(nx^{n-1}\). Hence, the derivative of \(f(x)\) is \(f^{\prime}(x) = \frac{1}{3}x^{-\frac{2}{3}}\).

Step 2: Find the second derivative

The second derivative is obtained by differentiating the first derivative. Differentiating \( f^{\prime}(x) = \frac{1}{3}x^{-\frac{2}{3}}\) again using the power rule gives \( f^{\prime\prime}(x) = -\frac{2}{9}x^{-\frac{5}{3}}\).

Step 3: Find the inflection point

The inflection point is where the second derivative is equal to zero or does not exist. But, since \(f^{\prime\prime}(x) = -\frac{2}{9}x^{-\frac{5}{3}}\) is never equal to zero and is not defined at \(x = 0\). Therefore, the function has an inflection point at \(x = 0\).

Step 4: Check if the second derivative exists at the inflection point

The second derivative \(f^{\prime\prime}(x) = -\frac{2}{9}x^{-\frac{5}{3}}\) does not exist when \(x = 0\), because this leads to division by zero, which is undefined. Hence, it can be confirmed that \(f^{\prime\prime}(x)\) does not exist at \(x = 0\)