Question

Find the maximum and minimum value of $\frac{1}{\cos ec\theta }$

Answer 1

Given that:

$f\left(\theta \right)=\frac{1}{cosec\theta }=sin\theta$

$D.w.rto\theta$ of $f\left(\theta \right)$, we get

$f^{\prime }\left(\theta \right)=cos\theta$

To get the minimum or maximum value,

$f^{\prime }\left(\theta \right)=0$

$cos\theta =0\Rightarrow cos\theta =cos{90}^{\circ }$

$\theta ={90}^{\circ }.$ ......(1)

Now,

$D.w.rto\theta$ of $f^{\prime }\left(\theta \right)$, we get

$f^{\prime \prime }\left(\theta \right)=-sin\theta =-sin{90}^{\circ }=-1$ [from equation (1), substituting the value of $\theta$]

since, $f^{\prime \prime }\left(\theta \right)$ is $negative$.

Hence, $f\left(\theta \right)$ is maximum at $\theta ={90}^{\circ }$.

To get the maximum value of the $f\left(\theta \right)=sin\theta ,$

$f\left({90}^{c}irc\right)=sin{90}^{\circ }=1.$ [On substituting the value of $\theta$]