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The values of a for which the function f (x) = sinx - ax + b increases on R are ______.

Answers (1)

Given f (x) = sinx - ax + b

Apply first derivative and get

\mathrm{f}^{\prime}(\mathrm{x})=\frac{\mathrm{d}(\sin \mathrm{x}-\mathrm{ax}+\mathrm{b})}{\mathrm{dx}}

Apply sum rule and 0 is the differentiation of the constant term, so
\mathrm{f}^{\prime}(\mathrm{x})=\frac{\mathrm{d}(\sin \mathrm{x})}{\mathrm{dx}}-\mathrm{a} \frac{\mathrm{d}(\mathrm{x})}{\mathrm{dx}}+0
Apply first derivative and get
f^{\prime}(x)=\cos x-a
Also  f(x) increases on R
\\ \Rightarrow \mathrm{f}^{\prime}(x) \geq 0 \ \ \forall x \in \mathbb{R} \\ \Rightarrow \cos x - a \geq 0 \ \ \forall x \in \mathbb{R} \\ \Rightarrow \cos x \geq a \ \ \forall x \in \mathbb{R}
This is possible when a\leq-1
Hence a \in(-\infty,-1]
The values of a increases on \mathrm{R}$ are $(-\infty,-1] .

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