Ling Xiang Ning of Raffles Institution in
Singapore, found two solutions, and after trying some more values
of x, decided that there would be no more.
Vassil Vassilev, who is in year 11 at
Lawnswood High School in Leeds, used a graph to convince himself
that there were no more solutions. First of all, he
rearranged the equation:
$$ \eqalign {x &=& (\sqrt2)^x & \\
\left(\sqrt[x]{x}\right) &=& \sqrt2 (taking the xth
root)\\ \left(\sqrt[x]{x}\right)^2 &=& 2 (squaring)
x^{\frac{2}{x}} &=& 2 }$$
Next he calculated values of x (2/x) for values of x
from 1 to 8, and plotted this graph:
Vassil commented that by looking at the original equation we
could rule out negative values of x, and that the values in the
graph decline after x=4. The justification for the decline is
that we are calculating smaller and smaller powers. However, you
may not be convinced, as the number we are finding powers
of is getting bigger. Are you sure that curve isn't
going to go up again further along?
The use of a graph to justify there being only 2 solutions was
a good idea.
Clearly x=0 is not a solution and there are no negative solutions
because the right hand side is positive. We have found 2 solutions
x=2 and x=4 and we have to show that there are no other solutions.
Consider the function f(x) = (Ö2)x = exp(xlnÖ2). Differentiating
the function gives:
|
|
df
dx
|
= (ln Ö2)×exp(xlnÖ2) |
|
and
|
|
d2f
dx2
|
= (ln Ö2)2×exp(xlnÖ2) > 0. |
|
So the graph of f is convex (the first derivative, or gradient, is
always increasing) so the graph of y=f(x) meets y=x in at most 2 points.
So 2 and 4 are the only solutions.