Question
Find the values of 'a' for which hte function $\text{f}(\text{x})=\sin\text{x}-\text{a}\text{x}+4$ is increasing function on R.
✓
Answer
$\text{f}(\text{x})=\sin\text{x}-\text{a}\text{x}+4$
$\text{f}'(\text{x})=\cos\text{x}-\text{a}$
Given: f(x) is increasing on R.
$\Rightarrow\text{f}'>\cos\text{x}-\text{a}$
$\Rightarrow\cos\text{x}>\text{a}$
We know,
$\cos\text{x}\geq-1,\forall\ \text{x}\in\text{R}$
$\Rightarrow\text{a}<-1$
$\Rightarrow\text{a}\in(-\infty,-1)$
$\text{f}'(\text{x})=\cos\text{x}-\text{a}$
Given: f(x) is increasing on R.
$\Rightarrow\text{f}'>\cos\text{x}-\text{a}$
$\Rightarrow\cos\text{x}>\text{a}$
We know,
$\cos\text{x}\geq-1,\forall\ \text{x}\in\text{R}$
$\Rightarrow\text{a}<-1$
$\Rightarrow\text{a}\in(-\infty,-1)$
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