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Graeme showed the inequalities for
us:
$7 = 9^{1/2} + 8^{1/3} + 16^{1/4} > 7^{1/2} + 7^{1/3} +
7^{1/4}$
$4 = 4^{1/2} + 1^{1/3} + 1^{1/4} < 4^{1/2} + 4^{1/3} +
4^{1/4}$
While I was at it, I came up with these:
$6 = 6.25^{1/2} + 6.859^{1/3} + 6.5536^{1/4} > 6^{1/2} + 6^{1/3}
+ 6^{1/4}$
Although that looks hard, it can be done without a calculator by
partitioning $6$ into $2.5+1.9+1.6$, and finding appropriate powers
of each number. The last one is easy for computer geeks like me who
have memorized many small powers of $2$.
$5 < 4^{1/2} + 4.096^{1/3} + 4^{1/4} < 5^{1/2} + 5^{1/3} +
5^{1/4}$
This, too, is pretty easy without a calculator - $4.096^{1/3}$ is
$1.6$, and the square root of $2$ is more than $1.4$, so the first
sum is more than $5$, and clearly less than the second sum.
Thanks for the extensions, Graeme.
These inequalities show that the graph is
going to intersect the x-axis somewhere between 4 and 7, which it
does: