Find the critical points of the function y = x-cos x.

1. Let’s calculate the derivative of the function:

y (x) = x – cosx;

y ‘(x) = 1 + sinx.

2. The derivative of a continuous and smooth function at critical points is equal to zero:

y ‘(x) = 0;

1 + sinx = 0;

sinx = -1;

x = -π / 2 + 2πk, k ∈ Z.

3. On the entire set of real numbers, the derivative of the function is greater than or equal to zero:

sinx ≥ -1;

sinx + 1 ≥ 0;

y ‘(x) ≥ 0,

hence, the function increases and the critical points are not extremum points.

Answer: critical points of the function: -π / 2 + 2πk, k ∈ Z.

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