Question

Minimum value of $f\left(x\right)=256{\sin }^{2}x+324cose{c}^{2}x$. for all of x $ϵ$ R is

• A. $576$
• B. $580$
• C. $584$
• D. None of these

Answer 1

The correct option is B $580$

$256{\sin }^{2}x+324{\csc }^{2}x={(16\sin x+18\csc x)}^2-\mathrm{2.16.18}$

differentiating given eq'n,we get $\frac{dy}{dx}=\frac{2cotx}{si{n}^{2}x}(256si{n}^{4}x-324)$

$\frac{dy}{dx}=0,{\sin }^{2}x=\frac{18}{16}\ge 1$ so not possible.

$(256si{n}^{4}x-324)$ is always $\le 0$ since $si{n}^{4}x\in (0,1)$

in the interval $x\in (0,\pi /2),\cot x\ge 0$

So, $\frac{dy}{dx}\le 0$ that means the function is decreasing in this interval.

in the interval, $(\pi /2,\pi ),\cot x\le 0.\frac{dy}{dx}\ge 0$ that means the function is increasing in this interval.

So, there is a minima at $x=\pi /2.$

at $x=\pi /2,y=256+324=580$

the minimum value of the given function is $580$