Question

If $a=(100{)}^{1/100}$ and $b=(101{)}^{1/101}$ then

• A. $a=b$
• B. $a>b$
• C. $a<b$
• D. none of these

Answer 1

Answer: (B)

Assume $f(x)=x^{1/x}$ and let us examine monotonic nature of $f(x)$
$f^{\prime }(x)=x^{1/x}\cdot \left(\frac{1-\ell nx}{x^2}\right)$
$f^{\prime }(x)>0\Rightarrow x\in (0,e)$
and $f^{\prime }(x)<0\Rightarrow (e,\infty )$
Hence $f(x)$ is M.D. for $x\ge e$

and since $100<101$
$\Rightarrow f(100)>f(101)$
$\Rightarrow (100{)}^{1/100}>(101{)}^{1/101}$