 Solution

Put $q=\lambda a+(1-\lambda )b$. Since $0<\lambda <1$ and $a<b$, $a<q<b$. Apply the mean value theorem separately to $[a,q]$ and $[q,b]$. There exist $x\in (a,q)$ and $y\in (q,b)$ such thatThese are the requested points, and $x<y$. Part (i) makes $f^{\prime }(x)\le f^{\prime }(y)$, so the corresponding secant slopes satisfyMultiply by the positive denominators and collect $f(q)$ to obtainThis is precisely the defining chord inequality for a convex function. The proof uses only existence and nonnegativity of $f^{\prime \prime }$, with no additional regularity hypothesis on that derivative.