Question

Let $M$ denote the maximum value of $f\left(x\right)={\sin }^{2}x-4\sin x+10$ and $m$ denote the minimum value of $g\left(x\right)=4{\sec }^{2}x+36{\text{cosec}}^{2}x-14.$ Then the value of $(m-M)$ is

Answer 1

$f\left(x\right)={\sin }^{2}x-4\sin x+10\Rightarrow f\left(x\right)=(\sin x-2{)}^2+6$
As $\sin x\in [-1,1]$, so the maximum value of $f$ occurs at $\sin x=-1$
$M=15$

$g\left(x\right)=4{\sec }^{2}x+36{\text{cosec}}^{2}x-14\Rightarrow g\left(x\right)=4+4{\tan }^{2}x+36+36{\cot }^{2}x-14\Rightarrow g\left(x\right)=4{\tan }^{2}x+36{\cot }^{2}x+26$

Applying A.M. $\ge$ G.M., we get
$4{\tan }^{2}x+36{\cot }^{2}x\ge 2\sqrt{4\times 36}\Rightarrow 4{\tan }^{2}x+36{\cot }^{2}x\ge 24$
So, the minimum value of $g$ is $m=24+26=50$

Hence, $m-M=50-15=35$