Find the \( x \) and \( y \) coordinates of all inflection points. \( f(x)=4 x^{\frac{5}{3}}+5 \) What is/are the inflection point(s)? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The inflection point(s) is/are (Type an ordered pair. Use a comma to separate answers as needed.) B. There are no inflection points.

We start with the function \[ f(x)=4x^{\frac{5}{3}}+5. \] **Step 1. Compute the first derivative.** Differentiate \( f(x) \) with respect to \( x \): \[ f'(x)=4\cdot\frac{5}{3}x^{\frac{5}{3}-1}=\frac{20}{3}x^{\frac{2}{3}}. \] **Step 2. Compute the second derivative.** Differentiate \( f'(x) \): \[ f''(x)=\frac{20}{3}\cdot\frac{2}{3}x^{\frac{2}{3}-1}=\frac{40}{9}x^{-\frac{1}{3}}. \] **Step 3. Determine where the concavity changes.** Inflection points occur where the concavity changes, i.e., where \( f''(x) \) changes sign. Note that \( f''(x) \) is not defined at \( x=0 \) because \( x^{-\frac{1}{3}}=\frac{1}{\sqrt[3]{x}} \) is undefined there. Thus, we check the sign of \( f''(x) \) around \( x=0 \): - For \( x>0 \), \( x^{-\frac{1}{3}}>0 \), so \[ f''(x)=\frac{40}{9}x^{-\frac{1}{3}}>0, \] meaning the graph is concave up. - For \( x<0 \), note that the cube root of a negative number is negative, so \( x^{-\frac{1}{3}} \) is negative, hence \[ f''(x)=\frac{40}{9}x^{-\frac{1}{3}}<0, \] meaning the graph is concave down. Since the concavity changes from down to up as \( x \) passes through \(0\), there is an inflection point at \( x=0 \). **Step 4. Find the \( y \)-coordinate of the inflection point.** Evaluate \( f(0) \): \[ f(0)=4\cdot0^{\frac{5}{3}}+5=0+5=5. \] **Conclusion:** The inflection point of the function is \[ (0,5). \]