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Increasing and Decreasing Functions — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIncreasing and Decreasing Functions2 Marks
Question
Find the values of b for which the function $\text{f}(\text{x})=\sin\text{x}-\text{b}\text{x}+\text{c}$ is a decreasing function on R.
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Answer

$\text{f}(\text{x})=\sin\text{x}-\text{b}\text{x}+\text{c}$$\text{f}'(\text{x})=\cos\text{x}-\text{b}$
Given: f(x) is decreasing on R. $\Rightarrow\text{f}'(\text{x})<0\ \forall\ \text{x}\in\text{R}$ $\Rightarrow\cos\text{x}-\text{b}<0\ \forall\ \text{x}\in\text{R}$ $\Rightarrow\cos\text{x}<\text{b},\forall\ \text{x}\in\text{R}$ $\Rightarrow\text{b}\geq1$ $[\because\ -1\leq\cos\text{x}\leq1]$ 

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