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 soparate answers as needed.) B. There are no inflection points.

1. Write the function: \[ f(x)=4x^{\frac{5}{3}}+5 \] 2. Find the first derivative \( f'(x) \): \[ f'(x)=4\cdot\frac{5}{3}x^{\frac{5}{3}-1}=\frac{20}{3}x^{\frac{2}{3}} \] 3. Compute the second derivative \( f''(x) \): \[ f''(x)=\frac{20}{3}\cdot\frac{2}{3}x^{\frac{2}{3}-1}=\frac{40}{9}x^{-\frac{1}{3}}=\frac{40}{9x^{\frac{1}{3}}} \] 4. Determine where \( f''(x) \) is zero or undefined. The expression \( f''(x)=\frac{40}{9x^{\frac{1}{3}}} \) is undefined at \( x=0 \). 5. Check the sign of \( f''(x) \) around \( x=0 \): - For \( x>0 \), \( x^{\frac{1}{3}}>0 \) so \( f''(x)>0 \). - For \( x<0 \), \( x^{\frac{1}{3}}<0 \) (since the cube root of a negative number is negative) so \( f''(x)<0 \). The sign of \( f''(x) \) changes at \( x=0 \), indicating an inflection point. 6. Find the \( y \)-coordinate of the inflection point by evaluating \( f(0) \): \[ f(0)=4\cdot0^{\frac{5}{3}}+5=5 \] 7. Therefore, the inflection point is: \[ (0,5) \] Answer choice A is correct with the inflection point \((0,5)\).