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## Main Question or Discussion Point

x in (0,pi];

f(x) = sin(x)-x

f'(x) = cos(x) - 2x;

f'(x) = 0 ==> cos(x) - 2x = 0;

since |cos(x)| ≤ 1,

cos(x) - 2x ≤ 1 - 2x;

Now 1-2x = 0 <==> x = 1/2;

f'(1/4) = cos(1/4) - 2*(1/4) > 0 and f'(3/4) = cos(3/4) - 2*(3/4) < 0;

==> x = 1/2 is maximum and f'(x) ≤ 1/2;

Is my logic correct?

f(x) = sin(x)-x

^{2};f'(x) = cos(x) - 2x;

f'(x) = 0 ==> cos(x) - 2x = 0;

since |cos(x)| ≤ 1,

cos(x) - 2x ≤ 1 - 2x;

Now 1-2x = 0 <==> x = 1/2;

f'(1/4) = cos(1/4) - 2*(1/4) > 0 and f'(3/4) = cos(3/4) - 2*(3/4) < 0;

==> x = 1/2 is maximum and f'(x) ≤ 1/2;

Is my logic correct?