Identify the coordinates of any local and absolute extreme points and inflection points. \[ y=6 x+6 \boldsymbol{\operatorname { s i n }} x, 0 \leq x \leq 2 \pi \]

To find the local and absolute extreme points and inflection points of the function \[ y = 6x + 6 \sin x \] on the interval \(0 \leq x \leq 2\pi\), we will follow these steps: 1. **Find the first derivative** to locate critical points. 2. **Determine the second derivative** to find inflection points. 3. **Evaluate the function at critical points and endpoints** to find absolute extrema. ### Step 1: Find the first derivative The first derivative of \(y\) is: \[ y' = \frac{d}{dx}(6x + 6 \sin x) = 6 + 6 \cos x \] To find critical points, we set the first derivative equal to zero: \[ 6 + 6 \cos x = 0 \] This simplifies to: \[ \cos x = -1 \] The solutions to \(\cos x = -1\) in the interval \(0 \leq x \leq 2\pi\) are: \[ x = \pi \] ### Step 2: Evaluate the function at critical points and endpoints Next, we evaluate \(y\) at the critical point \(x = \pi\) and at the endpoints \(x = 0\) and \(x = 2\pi\): - At \(x = 0\): \[ y(0) = 6(0) + 6 \sin(0) = 0 \] - At \(x = \pi\): \[ y(\pi) = 6(\pi) + 6 \sin(\pi) = 6\pi + 0 = 6\pi \] - At \(x = 2\pi\): \[ y(2\pi) = 6(2\pi) + 6 \sin(2\pi) = 12\pi + 0 = 12\pi \] ### Step 3: Determine absolute extrema Now we compare the values: - \(y(0) = 0\) - \(y(\pi) = 6\pi\) - \(y(2\pi) = 12\pi\) From these evaluations, we can conclude: - The **absolute minimum** occurs at \(x = 0\) with \(y(0) = 0\). - The **absolute maximum** occurs at \(x = 2\pi\) with \(y(2\pi) = 12\pi\). ### Step 4: Find the second derivative for inflection points Now, we find the second derivative: \[ y'' = \frac{d}{dx}(6 + 6 \cos x) = -6 \sin x \] To find inflection points, we set the second derivative equal to zero: \[ -6 \sin x = 0 \] This gives: \[ \sin x = 0 \] The solutions to \(\sin x = 0\) in the interval \(0 \leq x \leq 2\pi\) are: \[ x = 0, \pi, 2\pi \] ### Summary of Results - **Local and Absolute Extreme Points**: - Absolute minimum at \((0, 0)\) - Absolute maximum at \((2\pi, 12\pi)\) - **Inflection Points**: - Inflection points at \(x = 0\), \(x = \pi\), and \(x = 2\pi\) (note that these points are also endpoints). Thus, the coordinates of the extreme points and inflection points are: - Absolute minimum: \((0, 0)\) - Absolute maximum: \((2\pi, 12\pi)\) - Inflection points: \((0, 0)\), \((\pi, 6\pi)\), \((2\pi, 12\pi)\)