1. Let \( f(x)=x^{2}-8 \ln x \). Which of the following is the root interval for \( f(x) \) ? \( \begin{array}{llll}\text { a. }(0,1) & \text { b. }(4,5) & \text { c. }(2,3) & \text { d. }(3,4)\end{array} \)

To find the root interval for the function \( f(x)=x^{2}-8 \ln x \), we can evaluate the function at the endpoints of the given options. 1. For option (0,1): - \( f(0.5) = (0.5)^2 - 8 \ln(0.5) \) gives a positive value, indicating no root here. 2. For option (4,5): - \( f(4) = 16 - 8 \ln(4) \) - \( f(5) = 25 - 8 \ln(5) \) - Calculate to see if the signs change. 3. For option (2,3): - \( f(2) = 4 - 8 \ln(2) \) and \( f(3) = 9 - 8 \ln(3) \). 4. For option (3,4): - Again, check the values of \( f(3) \) and \( f(4) \). By evaluating these, you can see whether the signs of \( f(x) \) change, which indicates a root. You will find that the interval (3,4) has a sign change and is the root interval for \( f(x) \). Therefore, the answer is (d) (3,4).