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# Show that the function given by $f (x) = \sin \;x$ is strictly increasing in $\left(0, \: \large {\frac{\pi}{2}}\right)$

This is (a) part of the multi-part question q3

Toolbox:
• A function $f(x)$ is said to be a strictly increasing function on $(a,b)$ if $x_1 < x_2\Rightarrow f(x_1) < f(x_2)$ for all $x_1,x_2\in (a,b)$
• If $x_1 < x_2\Rightarrow f(x_1) > f(x_2)$ for all $x_1,x_2\in (a,b)$ then $f(x)$ is said to be strictly decreasing on $(a,b)$
• A function $f(x)$ is said to be increasing on $[a,b]$ if it is increasing (decreasing) on $(a,b)$ and it is increasing (decreasing) at $x=a$ and $x=b$.
• The necessary sufficient condition for a differentiable function defined on $(a,b)$ to be strictly increasing on $(a,b)$ is that $f'(x) > 0$ for all $x\in (a,b)$
• The necessary sufficient condition for a differentiable function defined on $(a,b)$ to be strictly decreasing on $(a,b)$ is that $f'(x) < 0$ for all $x\in (a,b)$
Given :
$f(x)=\sin x$
Differentiating w.r.t $x$ we get,
$f'(x)=\cos x$
The interval $(0,\large\frac{\pi}{2})$ belongs to the first quadrant.
$f(x)$ is strictly increasing in $(0,\large\frac{\pi}{2})$