Question

Identify which is greater : $\frac{1+e^{\frac{2}{3}}}{e^{\frac{1}{3}}}or\frac{1+{\pi }^{\frac{2}{3}}}{{\pi }^{\frac{1}{3}}}$

• A. $\frac{1+{\pi }^{\frac{2}{3}}}{{\pi }^{\frac{1}{3}}}$
• B. $\frac{1+e^{\frac{2}{3}}}{e^{\frac{1}{3}}}$
• C. Both are equal
  
• D. None of these
  

Answer 1

The correct option is A $\frac{1+{\pi }^{\frac{2}{3}}}{{\pi }^{\frac{1}{3}}}$

Let
$f\left(x\right)=\frac{1+x^{\frac{2}{3}}}{x^{\frac{1}{3}}}$
$=x^{\frac{-1}{3}}+x^{\frac{1}{3}}$
$f^{\prime }\left(x\right)$
$=\frac{-1}{3}{x}^{\frac{-4}{3}}+\frac{1}{3}{x}^{-\frac{2}{3}}$
$\ge 0$
Or
$x^{\frac{2}{3}}(1-x^{\frac{2}{3}})\ge 0$
Hence
$x\ge 1x\le 0$ or $x\ge 1$
Therefore $xϵ[-1,0]\cup [1,\infty )$ for f(x), to be an increasing function.
Now $e<\pi$
Hence
$f\left(e\right)<f\left(\pi \right)$.