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If f''(x) = sin(x), then f'(x) = -cos(x) + C.

If f'(0) = 6, then (by integration of f''(x)) -cos(0) + C = 6, so that C = 6 +cos(0) = 6 +1 = 7.

If f'(x) = -cos(x) + 7, then (by integration of f'(x)) f(x) = -sin(x) + 7x + D.

If f(0) = 2, then 2 = -sin(0) + 0 +D, so that

f(x) = -sin(x) + 7x + 2.

CHECK:

f(0) = 0 + 7*0 + 2, as expected

f'(x) = -cos(x) + 7 and f'(0) = -1 + 7 = 6 as expected

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