33 Prove that the function f(x)= cos x is :

Question:

33 Prove that the function $f(x)=\cos x$ is :

i. strictly decreasing on $(0, \pi)$

ii. strictly increasing in $(\pi, 2 \pi)$

iii. neither increasing nor decreasing in $(0,2 \pi)$

Solution:

Given $f(x)=\cos x$

$\therefore f^{\prime}(x)=-\sin x$

(i) Since for each $x \in(0, \pi), \sin x>0$

$\Rightarrow \therefore f^{\prime}(x)<0$

So $f$ is strictly decreasing in $(0, \pi)$

(ii) Since for each $x \in(\pi, 2 \pi), \sin x<0$

$\Rightarrow \therefore f^{\prime}(x)>0$

So $f$ is strictly increasing in $(\pi, 2 \pi)$

(iii) Clearly from (1) and (2) above, $f$ is neither increasing nor decreasing in $(0,2 \pi)$

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