Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions3 Marks
Question
Write the interval in which $\text{f}(\text{x})=\sin\text{x}+\cos\text{x},\text{x}\in\Big[0,\frac{\pi}{2}\Big]$ is increasing.
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Answer
$\text{f}(\text{x})=\sin\text{x}+\cos\text{x},\text{x}\in\Big[0,\frac{\pi}{2}\Big]$$\text{f}'(\text{x})=\cos\text{x}-\sin\text{x}$
For f(x) to be increasing, we must have$\text{f}'(\text{x}) >0$
$\Rightarrow\cos\text{x}-\sin\text{x}>0$ $\Rightarrow\sin\text{x}<\cos\text{x}$ $\Rightarrow\frac{\sin\text{x}}{\cos\text{x}}<1$ $\Rightarrow\tan\text{x}<1$ $\Rightarrow\text{x}\in\Big[0,\frac{\pi}{4}\Big)$
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