Here is another approach.  Define a function $g$ that is equal to $f$ on $[0, 1/t]$, and on $[1/t, 1]$ is a linear interpolation between the points $(1/t, f(1/t))$ and $(1, 0)$.  Then we can check that $g$ is concave on all of $[0, 1]$ and $g \ge f$ on all of $[0, 1]$.  So now we can apply Jensen's inequality to $g$ and we're done.

I learned this method of "concave majorants" (or convex minorants) from Steele's book called *The Cauchy-Schwarz Master Class*.