For what value of lambda does the function f( | vedclass

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(B) function $f(x)$ increases monotonically if its derivative $f'(x) \ge 0$ for all $x \in \mathbb{R}$.Given $f(x) = \lambda x - \sin x$.Find the deri

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$\lambda > 1$

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(B) function $f(x)$ increases monotonically if its derivative $f'(x) \ge 0$ for all $x \in \mathbb{R}$.Given $f(x) = \lambda x - \sin x$.Find the derivative: $f'(x) = \lambda - \cos x$.For $f(x)$ to be monotonically increasing,we require $f'(x) \ge 0$,which means $\lambda - \cos x \ge 0$,or $\lambda \ge \cos x$.Since the maximum value of $\cos x$ is $1$,for the inequality $\lambda \ge \cos x$ to hold for all $x$,we must have $\lambda \ge 1$.Thus,the function increases monotonically when $\lambda \ge 1$.

For what value of $\lambda$ does the function $f(x) = \lambda x - \sin x$ increase monotonically?

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