Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions2 Marks
Question
Show that $\text{f}(\text{x})=\text{x}+\cos\text{x}-\text{a}$ is an increasing function on R for all values of a.
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Answer
We have, $\text{f}(\text{x})=\text{x}+\cos\text{x}-\text{a}$$\therefore\ \text{f}'(\text{x})=1-\sin\text{x}=\frac{2\cos^2\text{x}}{2}$
Now, $\text{x}\in\text{R}$ $\Rightarrow\frac{\cos^2\text{x}}{2}>0$ $\Rightarrow\frac{2\cos^2\text{x}}{2}>0$ $\Rightarrow\text{f}'(\text{x})>0$ Hence, f(x) is an increasing function for $\text{x}\in\text{R}.$
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