Get rid of your calculator 2

Calculus Level 3

Arrange the following in descending order : $\color{#20A900}{\sqrt [1]{1},\sqrt [2]{2},\sqrt [3]{3},\sqrt [4]{4},\sqrt [5]{5},\sqrt [6]{6},\sqrt [7]{7}}$

If the arrangement is of the form $\sqrt [n_{1}]{n_{1}},\sqrt [n_{2}]{n_{2}},\ldots\sqrt [n_{7}]{n_{7}}$ ,find the value of $\overline {n_{1}n_{2}n_{3}\ldots n_{7}}$ .

in case of $\sqrt[4]{4},\sqrt [2]{2}$ ,enter $4$ first and then $2$ .i.e. $\overline {\ldots42\ldots}$ .
This problem is created by me.
Have you tried the first part ?

The answer is 3425671.

1 solution

Gabriel Vasto
Jan 24, 2019

If we define a function, f(x)=x^(1/x), we can show that f'(x)>0 only if x<e, i.e., f does have a maximum at x=e. Then, for values of x such that x<e, f will be a crescent function, and for x>e, a decrescent one. Hence, we already know that 3>4>5>6>7 (and, of course, all of this must be >1). But notice that the fourth root of 4 is exactly the same as the second root of 2, which leads us to the corret order: 3425671.

Get rid of your calculator 2

The answer is 3425671.

1 solution

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