= Solution
The largest angular-frequency magnitude is $\omega_3=\sqrt{(2+\sqrt2)k/m}$. With overall amplitude $A$, the corresponding displacements are
$$
\boxed{x_1=x_3=\frac A2\cos(\omega_3t),\qquad
x_2=-\frac A{\sqrt2}\cos(\omega_3t)}.
$$
Thus the two outer masses move together, while the middle <mass> moves oppositely with $\sqrt2$ times their amplitude. All three have the same period $2\pi/\omega_3$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-2-highest-spring-mode.png]
{title=Highest-frequency mode of three equal masses between fixed supports}
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