Problem
8. Find the value of the constant $b$ such that the following function has a point of inflection at $x=3$. $f(x)=\sqrt{x+1}+\frac{b}{x}$
Solution
Step 1 ：Find the second derivative of the function $f(x) = \sqrt{x+1} + \frac{b}{x}$.
Step 2 ：Calculate the second derivative at $x=3$ and set it equal to 0.
Step 3 ：Solve the equation $\frac{2b}{27} - \frac{1}{32} = 0$ for the value of $b$.
Step 4 ：\(\boxed{\frac{27}{64}}\) is the value of the constant $b$ such that the function has a point of inflection at $x=3$.
