Question:
Mark the correct alternative in the following:
The function $f(x)=-\frac{x}{2}+\sin x$ defined on $\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$ is
A. increasing
B. decreasing
C. constant
D. none of these
Solution:
Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$
Given:-
$f(x)=-\frac{x}{2}+\sin x$
$\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=-\frac{1}{2}+\cos \mathrm{x}=\mathrm{f}^{\prime}(\mathrm{x})$
checking the value of $x$
$\cos -\frac{1}{2}>0$
hence increasing
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