If you know that \|f''\|_\infty then you should have a bound on \|f'\|_\infty in the same way that a bound on \|f'\|_\infty leads to a bound on $\|f\|_\infty$ (given some information on $f$ -- so we will look at this).
I am assuming that $f$ is differentiable on $[0,1]$ (twice differentiable on $[0,1]$ in the sequel). You can show that if \|f'\|_\infty then $f(a)+L$ is an upperbound for $f$ on $[0,1]$ (for a proof, suppose that $f(x)>f(0)+L$ for some $x\in(0,1]$ and show that this contradicts the Mean Value Theorem).
We can probably adapt this to your problem where f' plays the role of $f$ and f'' the role of f'. Using it straight away we have
\|f'\|_\infty\leq f'(0)+K.
Now we don't know f'(0) but I suppose but we could just use any f'(x) for $x\in(0,1)$ if we got one. We can do this using the Mean Value Theorem if we know that $f(a)=f_u(x)$ and $f(b)=f_l(x)$ (or we could probably do with estimates --- let's run with estimates).
Suppose that we have $m\leq f(x)\leq M$ (just take $m=f_l(x)$ and $M=\max\{|f_u(x)|,|f_l(x)|\}$.)
Now, can we say anything about
$\frac{M-m}{x_M-x_m},$
where $x_M$, $x_m$ are points that are 'close' to the maxima: $f(x_m)\sim f_m(x)$ and $f(x_M)\sim f_M(x)$. Well we have a problem if the max and min are close together...
I'm stuck at this point but I hope this hint helps --- any more information on the problem (how far apart are the maxima/ minima found, is there a local max/ min inside $[0,1]$ so that we could take f'(a)=0 there and work from this? Two pieces of information about $f$ or two about f' will do.) and we might be able to take these problems to fruition.