Question

Let $f(x)=1+3x^2+5x^4+7x^6+\dots \dots +21\cdot x^{20},x\in R$ and $g(x)=-x^2+4{\cos }^2\theta -4\sin \theta -7,\theta \in R$ If $d$ is the shortest distance between $f(x)\&g(x)$ and $d_1,d_2$ are the least and greatest value of $d$ respectively, then find $\left(\frac{d_2}{d_1}\right)$.

Answer 1

Answer: 4

$f(x)=1+3x^2+5x^4+7x^6+\dots \dots +21x^{20},x\in R$
$g(x)=-x^2+4{\cos }^2\theta -4\sin \theta -7,\theta \in R$
${f(x)|}_{\min }=1,{g(x)|}_{\max }=4{\cos }^2\theta -4\sin \theta -7\text{ at }x=0$
$\text{ shortest distance between }f(x)\text{ and }g(x)\text{ is }$
$d=1-\left(4{\cos }^2\theta -4\sin \theta -7\right)$
$=1-\left(4-4{\sin }^2\theta -4\sin \theta -7\right)$
$=4{\sin }^2\theta +4\sin \theta +4$
$=(2\sin \theta +1{)}^2+3$
$d_1=3,d_2=12$
$\therefore \frac{d_2}{d_1}=4$