Determine the absolute maximum and absolute minimum of
y = x - cosx
on the interval [0, 2pi]
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To find the absolute maximum and minimum of the function y = x - cosx on the interval [0, 2π], we need to evaluate the function at both the critical points and the endpoints of the interval.
First, we need to find the critical points by taking the derivative of the function with respect to x:
dy/dx = 1 + sinx.
Setting this equal to zero, we get:
1 + sinx = 0.
Solving for x, we find that x = π.
Next, we evaluate the function at the critical point x = π:
y(π) = π - cosπ = π + 1.
Now, we evaluate the function at the endpoints of the interval [0, 2π]:
y(0) = 0 - cos0 = 0 - 1 = -1,
y(2π) = 2π - cos(2π) = 2π - 1.
Comparing these values, we see that the absolute maximum of y = x - cosx on the interval [0, 2π] is 2π - 1, which occurs at x = 2π, and the absolute minimum is -1, which occurs at x = 0.
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