Final answer:
To identify the local maximum points of the function y = 4x + 4√2cos(x), first calculate the first and second derivatives. Then find the critical points by setting the first derivative equal to zero. Evaluate the second derivative at each critical point to determine if it is a local maximum.
Step-by-step explanation:
To identify the coordinates of the local and absolute extreme points and inflection points, we need to calculate the first derivative and second derivative of the function and solve for critical points. The local maximum points will occur at these critical points. The function is y = 4x + 4√2cos(x), 0≤x≤2π.
1. Find the first derivative of the function: y'(x) = 4 - 4√2sin(x).
2. Find the second derivative of the function: y''(x) = -4√2cos(x).
3. Set y'(x) = 0 and solve for critical points. These will be the x-coordinates of any local maximum or minimum points.
4. Evaluate y''(x) at each critical point to determine whether the point is a local maximum or minimum. If y''(x) > 0, it is a local minimum. If y''(x) < 0, it is a local maximum.
5. Plot the graph of the function y = 4x + 4√2cos(x) on the interval 0≤x≤2π, and indicate the coordinates of the local maximum points.