  We develop a variational approach to calculate the density response function
at finite temperatures of the lowest-lying two-fluid modes in a trapped
two-component Fermi superfluid close to a Feshbach resonance. The out-of-phase
oscillations, which are the analogue in trapped gases of second sound in
uniform superfluids, have so far not been observed in cold-atom experiments. At
unitarity, we show that these modes are observable at finite temperatures via
two-photon Bragg scattering, whose spectrum is related to the imaginary part of
density response function. This provides direct evidence for superfluidity and
a promising way to test microscopic results for thermodynamics at unitarity.

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