Find the Taylor polynomial of order 3 at \( a=\frac{\pi}{8} \) A. None of the options B. \( P_{3}, \frac{\pi}{8}(x)=1+4\left(x-\frac{\pi}{8}\right)-8\left(x-\frac{\pi}{8}\right)^{2}+\frac{64}{3}\left(x-\frac{\pi}{8}\right)^{3} \) C. \( P_{3}, \frac{\pi}{8}(x)=1+4\left(x-\frac{\pi}{8}\right)+16\left(x-\frac{\pi}{8}\right)^{2}+\frac{\alpha 4}{6}\left(x-\frac{\pi}{8}\right)^{3} \) D. \( P_{3}, \frac{\pi}{8}(x)=1+4\left(x-\frac{\pi}{8}\right)+8\left(x-\frac{\pi}{8}\right)^{2}+\frac{64}{3}\left(x-\frac{\pi}{8}\right)^{3} \) E. \( P_{3}, \frac{\pi}{8}(x)=1-2\left(x-\frac{\pi}{8}\right)+6\left(x-\frac{\pi}{8}\right)^{2}-\frac{32}{3}\left(x-\frac{\pi}{8}\right)^{3} \)

To find the Taylor polynomial of order 3 for a function \( f(x) \) at \( a = \frac{\pi}{8} \), we need to compute the derivatives of \( f(x) \) at \( a \) and use the Taylor series expansion formula: \[ P_3(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 \] Let's assume \( f(x) \) is a function that we need to analyze. Since the function is not specified in the problem, we will need to derive the Taylor polynomial based on the options provided. ### Step 1: Identify the function and its derivatives We will analyze the options to see if we can identify a function \( f(x) \) that matches the Taylor polynomial forms given. ### Step 2: Analyze the options 1. **Option B**: \[ P_{3}, \frac{\pi}{8}(x)=1+4\left(x-\frac{\pi}{8}\right)-8\left(x-\frac{\pi}{8}\right)^{2}+\frac{64}{3}\left(x-\frac{\pi}{8}\right)^{3} \] - Here, \( f(a) = 1 \), \( f'(a) = 4 \), \( f''(a) = -16 \), \( f'''(a) = \frac{64}{3} \). 2. **Option C**: \[ P_{3}, \frac{\pi}{8}(x)=1+4\left(x-\frac{\pi}{8}\right)+16\left(x-\frac{\pi}{8}\right)^{2}+\frac{\alpha 4}{6}\left(x-\frac{\pi}{8}\right)^{3} \] - Here, \( f(a) = 1 \), \( f'(a) = 4 \), \( f''(a) = 32 \), \( f'''(a) = \frac{\alpha 4}{6} \). 3. **Option D**: \[ P_{3}, \frac{\pi}{8}(x)=1+4\left(x-\frac{\pi}{8}\right)+8\left(x-\frac{\pi}{8}\right)^{2}+\frac{64}{3}\left(x-\frac{\pi}{8}\right)^{3} \] - Here, \( f(a) = 1 \), \( f'(a) = 4 \), \( f''(a) = 16 \), \( f'''(a) = \frac{64}{3} \). 4. **Option E**: \[ P_{3}, \frac{\pi}{8}(x)=1-2\left(x-\frac{\pi}{8}\right)+6\left(x-\frac{\pi}{8}\right)^{2}-\frac{32}{3}\left(x-\frac{\pi}{8}\right)^{3} \] - Here, \( f(a) = 1 \), \( f'(a) = -2 \), \( f''(a) = 12 \), \( f'''(a) = -\frac{32}{3} \). ### Step 3: Compare derivatives To determine which option is correct, we need to know the function \( f(x) \) and its derivatives at \( a = \frac{\pi}{8} \). Assuming \( f(x) \) is a trigonometric or polynomial function, we can compute the derivatives at \( a = \frac{\pi}{8} \) and compare them with the values in the options. ### Conclusion Without the specific function \( f(x) \), we cannot definitively determine which option is correct. However, if we assume that the derivatives match a known function, we can select the option that corresponds to those derivatives. If you have a specific function in mind or additional context, please provide it, and I can help you find the correct Taylor polynomial. Otherwise, based on the analysis of the options, it seems that **Option D** is a plausible candidate, as it has consistent values for the derivatives. Thus, the answer is: **D. \( P_{3}, \frac{\pi}{8}(x)=1+4\left(x-\frac{\pi}{8}\right)+8\left(x-\frac{\pi}{8}\right)^{2}+\frac{64}{3}\left(x-\frac{\pi}{8}\right)^{3} \)**.