Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions2 Marks
Question
Show that $\text{f}(\text{x})=\text{x}^2-\text{x}\sin\text{x}$ is an increasing function on $\Big(0,\frac{\pi}{2}\Big).$
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Answer
We have, $\text{f}(\text{x})=\text{x}^2-\text{x}\sin\text{x}$ $\therefore\ \text{f}'(\text{x})=2\text{x}-\sin\text{x}-\text{x}\cos\text{x}$ Now, $\text{x}\in\Big(0,\frac{\pi}{2}\Big)$ $\Rightarrow0\leq\sin\text{x}\leq1,0\leq\cos\text{x}\leq1$ $\Rightarrow2\text{x}-\sin\text{x}-\text{x}\cos\text{x}>0$ $\Rightarrow\text{f}'(\text{x})\geq0$ So, f(x) is strictly increasing function on $\Big(0,\frac{\pi}{2}\Big).$
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