Question

The minimum integral value of the sum of real numbers $x^{11},x^3,x^7,1,4x^{-5}$ and $x^{-1}$, where x can take only those values for which $f\left(x\right)=ln\left(x\right)$ is well defined is -

• A. $8$
• B. $9$
• C. $10$
• D. $11$

Answer 1

The correct option is B $9$

We know,

$f\left(x\right)=ln\left(x\right)$ is well defined for $xϵ(0,\infty )$

If $\underset{–}{0<x\le 1}$

$\therefore x^{11},x^3,x^7$ values will be less than $x$

$\therefore x^{-5},x^{-1}$ values will be greater than $x$

If $\underset{–}{1\le x<\infty }$

$\therefore x^{11},x^3,x^7$ values will be greater than $x$

$\therefore x^{-5},x^{-1}$ values will be less than $x$

$\therefore$ Point of inflexion of the summation will be at $x=1$

$\therefore$ Minimum Sum $=1^{11}+1^3+1^7+1+4\times 1^{-5}+1^{-1}$

$=1+1+1+1+4+1$

$=9$