Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.) \[ f(x)=10 \csc \left(\frac{3 x}{2}\right),(0,2 \pi) \] \( (x, y)=( \) DNE Determine the open intervals on which the graph of the function Is concave upward or concave downward. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) concave upward concave downward \( \left(\frac{2 \pi}{3}, \frac{2 \pi}{3}\right) \)

To find the points of inflection and determine the concavity of the function \( f(x) = 10 \csc\left(\frac{3x}{2}\right) \), we need to follow these steps: 1. **Find the second derivative \( f''(x) \)**. 2. **Set \( f''(x) = 0 \) to find potential points of inflection**. 3. **Determine the sign of \( f''(x) \) to find intervals of concavity**. Let's start by calculating the first and second derivatives of the function. ### Step 1: First Derivative The first derivative \( f'(x) \) can be found using the derivative of the cosecant function. The derivative of \( \csc(u) \) is \( -\csc(u) \cot(u) \cdot u' \). Let \( u = \frac{3x}{2} \), then \( u' = \frac{3}{2} \). Thus, the first derivative is: \[ f'(x) = 10 \cdot -\csc\left(\frac{3x}{2}\right) \cot\left(\frac{3x}{2}\right) \cdot \frac{3}{2} \] ### Step 2: Second Derivative Now, we need to differentiate \( f'(x) \) to find \( f''(x) \). Let's calculate \( f'(x) \) first and then differentiate it to find \( f''(x) \). ### Step 3: Find Points of Inflection After finding \( f''(x) \), we will set it to zero and solve for \( x \). ### Step 4: Determine Concavity We will analyze the sign of \( f''(x) \) to determine the intervals of concavity. Let's perform the calculations for \( f'(x) \) and \( f''(x) \). Evaluate the derivative by following steps: - step0: Solution: \(\left(f\right)^{\prime}\times x\) - step1: Find the derivative: \(1\times x\) - step2: Multiply: \(x\) Calculate or simplify the expression \( 10 * -\csc(3x/2) * \cot(3x/2) * (3/2) \). Simplify the expression by following steps: - step0: Solution: \(10\left(-\csc\left(\frac{3x}{2}\right)\right)\cot\left(\frac{3x}{2}\right)\times \frac{3}{2}\) - step1: Rewrite the expression: \(10\left(-1\right)\csc\left(\frac{3x}{2}\right)\cot\left(\frac{3x}{2}\right)\times \frac{3}{2}\) - step2: Use the rules for multiplication and division: \(-10\csc\left(\frac{3x}{2}\right)\cot\left(\frac{3x}{2}\right)\times \frac{3}{2}\) - step3: Multiply the terms: \(-15\csc\left(\frac{3x}{2}\right)\cot\left(\frac{3x}{2}\right)\) - step4: Calculate: \(-15\times \frac{1}{\sin\left(\frac{3x}{2}\right)}\times \cot\left(\frac{3x}{2}\right)\) - step5: Calculate: \(-15\times \frac{1}{\sin\left(\frac{3x}{2}\right)}\times \frac{\cos\left(\frac{3x}{2}\right)}{\sin\left(\frac{3x}{2}\right)}\) - step6: Calculate: \(-\frac{15\cos\left(\frac{3x}{2}\right)}{\sin^{2}\left(\frac{3x}{2}\right)}\) - step7: Rewrite the expression: \(-15\sin^{-2}\left(\frac{3x}{2}\right)\cos\left(\frac{3x}{2}\right)\) - step8: Calculate: \(-15\left(1+\cot^{2}\left(\frac{3x}{2}\right)\right)\cos\left(\frac{3x}{2}\right)\) - step9: Calculate: \(\left(-15-15\cot^{2}\left(\frac{3x}{2}\right)\right)\cos\left(\frac{3x}{2}\right)\) - step10: Rewrite the expression: \(-15\cos\left(\frac{3x}{2}\right)-15\cos\left(\frac{3x}{2}\right)\cot^{2}\left(\frac{3x}{2}\right)\) - step11: Calculate: \(-15\cos\left(\frac{3x}{2}\right)-15\cos\left(\frac{3x}{2}\right)\left(\csc^{2}\left(\frac{3x}{2}\right)-1\right)\) - step12: Calculate: \(-15\csc^{2}\left(\frac{3x}{2}\right)\cos\left(\frac{3x}{2}\right)\) The first derivative of the function is: \[ f'(x) = -15 \csc^{2}\left(\frac{3x}{2}\right) \cos\left(\frac{3x}{2}\right) \] Now, let's find the second derivative \( f''(x) \). The result indicates that \( f''(x) = x \). ### Step 5: Find Points of Inflection To find the points of inflection, we set \( f''(x) = 0 \): \[ x = 0 \] ### Step 6: Determine Concavity Next, we need to analyze the sign of \( f''(x) \) around the point \( x = 0 \): - For \( x < 0 \), \( f''(x) < 0 \) (concave downward). - For \( x > 0 \), \( f''(x) > 0 \) (concave upward). ### Summary of Results 1. **Point of Inflection**: - \( (x, y) = (0, f(0)) \) - We need to calculate \( f(0) \): \[ f(0) = 10 \csc(0) \text{ (undefined)} \] Therefore, the point of inflection does not exist (DNE). 2. **Concave Upward**: - The function is concave upward for \( (0, 2\pi) \). 3. **Concave Downward**: - The function is concave downward for \( (0, 0) \) (not applicable since it's a single point). ### Final Answers - Point of Inflection: \( (x, y) = \text{DNE} \) - Concave Upward: \( (0, 2\pi) \) - Concave Downward: \( \text{DNE} \) (since it only changes at a single point).