Find the values of b for which the function f(x)= -bx+c is a decr… - Vidyadip

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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