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1.Relation and Function
normal
વિધેય $f(x) = {x^{10}} + {x^2} + \frac{1}{{{x^{12}}}} + \frac{1}{{\left( {1\ +\ {{\sec }^{ - 1}}\ x} \right)}}$ ની ન્યુનતમ કિમત ........ છે.
A
$\frac{{\pi\ +\ 4}}{{\pi\ +\ 1}}$
B
$\frac{{3\pi\ +\ 4}}{{\pi\ +\ 1}}$
C
$\frac{{\pi\ +\ 4}}{{3\pi\ +\ 1}}$
D
$3$
Solution
$x^{10}+x^{2}+\frac{1}{x^{12}} \geq 3$ $(AM.GM)$
Equality occurs of $x=\pm 1$
But ${\sec ^{ – 1}}{\rm{x}}$ takes maximum at $\mathrm{x}=-$
$f_{\min }(\mathrm{x})=f(-1)=3+\frac{1}{\pi+1}$
Standard 12
Mathematics
