To be concave on $[0,1]$, $f(t)\leq f(0)+tf'(0)$ is enough? Suppose $f(t)>0$.

$f(t)\leq f(0)+tf'(0)$ if and only if $f$ is concave over $[0,1]$.

Is the above stetement true? There must be a counterexample in my
opinion.

An obvious counterexample would be $f(t)=\sin(2\pi t)+1$, which is neither convex nor concave but fulfills your inequality.
