Without loss of generality we migth and shall assume $f'_{+}(a)>t>f'_{-}(b)$ Let $g(x):=f(x)-tx$ Then $g'(x)=f'(x)-t$ $g'_{+}(a)>0>g'_{-}(b)$ and we wish to find a zero of $g'$

Since $g$ is a continuous function on $[a,b]$ it attains a maximum on $[a,b]$ Since $g'_+(a)>0$ and $g'_+(b)<0$ Fermat's theorem states that neither $a$ nor $b$ can be points where $f$ has a local maximum. So a maximum is attained at some $c \in (a,b)$ But then $g'(c)=0$ again by Fermat's theorem.