Find the \( x \) and \( y \) coordinates of all inflection points. \( f(x)=5 x^{\frac{5}{3}}+6 \) What is/are the inflection poin'(s)? Selecl the correct choice below and, if necessary, fill in the answer box to complote your choice. A. The innection poinl(s) is/are \( \square \). (Type an ordered pair. Use a comma to separate answers as needed.) B. There are no inflection points.

To find the inflection points of the function \( f(x)=5 x^{\frac{5}{3}}+6 \), we first need to find the second derivative and determine where it changes sign. 1. Find the first derivative \( f'(x) \): \[ f'(x) = \frac{d}{dx}(5x^{\frac{5}{3}} + 6) = \frac{25}{3} x^{\frac{2}{3}}. \] 2. Find the second derivative \( f''(x) \): \[ f''(x) = \frac{d}{dx}\left(\frac{25}{3} x^{\frac{2}{3}}\right) = \frac{50}{9} x^{-\frac{1}{3}}. \] 3. Determine where \( f''(x) = 0 \) or is undefined. The second derivative is undefined at \( x = 0 \). 4. Analyze the sign of \( f''(x) \): - For \( x < 0 \), \( f''(x) < 0 \) (concave down). - For \( x > 0 \), \( f''(x) > 0 \) (concave up). Since the concavity changes at \( x = 0 \), we have an inflection point. Thus, the inflection point occurs at \( (0, f(0)) = (0, 6) \). A. The inflection point(s) is/are \( (0, 6) \).