Relationship between $n\geq 9$ & $\mu=n^{\frac{1}{2}}+ n^{\frac{1}{3}}+n^{\frac{1}{4}}$

Question

Suppose $n\geq 9$ is an integer. Let $\mu=n^{\frac{1}{2}}+ n^{\frac{1}{3}}+n^{\frac{1}{4}}$. Then which of the following relationships between $n$ and $μ$ is correct?

1)$n=\mu$

2)$n<\mu$

3)$n>\mu$

Answer 1

Firstly we can rule out option 1 since if we take $n = 9$ then $\mu = 3 + 9^\frac{1}{3} + 9^\frac{1}{4} < 9$

This also rules out option 2 by counterexample. Now we are left with option 3.

Now note that $n^\frac{1}{3}, n^\frac{1}{4} < n^\frac{1}{2}$

Therefore we can consider $n - 3*n^\frac{1}{2}$

For $n \geqslant 9$:

$3*n^\frac{1}{2} <n$

$\Rightarrow n > n^\frac{1}{2} + n^\frac{1}{3} + n^\frac{1}{4} = \mu$

Therefore option 3 is correct.