suppose by way of contradiction that f'(x) has two zeros in [a,b], say at x = c and x = d with c < d.

then at some point k in (c,d) (which is contained in [a,b]), by rolle's therorem we have f"(k) = 0, contradicting that f"(x) > 0 on [a,b].

the same proof works if f"(x) < 0 on [a,b].

it is important that f"(x) > 0 and not f"(x) ≥ 0, for we might have f(x) = c (a constant), in which case f'(x) has infinitely many roots on any interval.