Find the intervals on which the graph of \( f \) is concave upward, the intervals on which the graph of \( f \) is concave downward, and the inflection points. \( f(x)=\ln \left(x^{2}-2 x+10\right) \) For what interval(s) of \( x \) is the graph of \( f \) concave upward? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. B. The graph is never concave upward. For what interval(s) of \( x \) is the graph of \( f \) concave downward? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \square \) (Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as needed.) B. The graph is never concave downward. Determine the \( x \) coordinates of any inflection points of the graph of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( x=\square \) (Type an exact answer. Use a comma to separate answers as needed.) A comma to separate answers as needed.) A

To find the intervals on which the graph of \( f(x) = \ln(x^2 - 2x + 10) \) is concave upward, concave downward, and the inflection points, we need to find the second derivative of the function and analyze its sign. Given function: \( f(x) = \ln(x^2 - 2x + 10) \) First, let's find the first derivative of the function: \[ f'(x) = \frac{2x - 2}{x^2 - 2x + 10} \] Next, let's find the second derivative of the function: \[ f''(x) = \frac{(x^2 - 2x + 10)(2) - (2x - 2)(2x - 2)}{(x^2 - 2x + 10)^2} \] Simplifying the second derivative: \[ f''(x) = \frac{2x^2 - 4x + 20 - 4x^2 + 8x - 4}{(x^2 - 2x + 10)^2} \] \[ f''(x) = \frac{-2x^2 + 4x + 16}{(x^2 - 2x + 10)^2} \] Now, we need to find the intervals on which the graph of \( f \) is concave upward and concave downward by analyzing the sign of the second derivative. Concave upward: \( f''(x) > 0 \) Concave downward: \( f''(x) < 0 \) Let's analyze the sign of the second derivative to determine the intervals on which the graph of \( f \) is concave upward and concave downward. Solve the inequality by following steps: - step0: Solve the inequality by testing the values in the interval: \(-2x^{2}+4x+16>0\) - step1: Rewrite the expression: \(-2x^{2}+4x+16=0\) - step2: Factor the expression: \(-2\left(x-4\right)\left(x+2\right)=0\) - step3: Divide the terms: \(\left(x-4\right)\left(x+2\right)=0\) - step4: Separate into possible cases: \(\begin{align}&x-4=0\\&x+2=0\end{align}\) - step5: Solve the equation: \(\begin{align}&x=4\\&x=-2\end{align}\) - step6: Determine the test intervals: \(\begin{align}&x<-2\\&-24\end{align}\) - step7: Choose a value: \(\begin{align}&x_{1}=-3\\&x_{2}=1\\&x_{3}=5\end{align}\) - step8: Test the chosen value: \(\begin{align}&x<-2\textrm{ }\textrm{is not a solution}\\&-24\textrm{ }\textrm{is not a solution}\end{align}\) - step9: The final solution is \(-2