The function sin x - bx + c will be increasin | vedclass

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(C) Let $f(x) = \sin x - bx + c$.For the function to be increasing on $(-\infty, \infty)$,we must have $f'(x) \ge 0$ for all $x \in \mathbb{R}$.Calcul

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$b \le -1$

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(C) Let $f(x) = \sin x - bx + c$.For the function to be increasing on $(-\infty, \infty)$,we must have $f'(x) \ge 0$ for all $x \in \mathbb{R}$.Calculating the derivative,we get $f'(x) = \cos x - b$.Setting $f'(x) \ge 0$,we have $\cos x - b \ge 0$,which implies $\cos x \ge b$.Since the range of $\cos x$ is $[-1, 1]$,the inequality $\cos x \ge b$ holds for all $x$ if and only if the minimum value of $\cos x$ is greater than or equal to $b$.The minimum value of $\cos x$ is $-1$.Therefore,we must have $-1 \ge b$,or $b \le -1$.

The function $\sin x - bx + c$ will be increasing in the interval $(-\infty, \infty)$,if

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