How to get a approx nature of $f'''(x)$ graph from the given data ( Graph techinque using LMVT etc., or using just calculus both suffices ) The above mentioned title is for giving a rough graph for $f'''(x)$ , such that we can show that there exists some values of $f'''(c) \ge 6$. Problem where i encountered this:  " If $f(x)$ is a function with a continuous third derivative on $[0, 2]$ such that
$f(0) = f'(0) = f''(0) = f'(2) = f''(2) = 0$ and $f(2) = 2$, then minimum number of value(s) of $c$ belongs $[0, 2]$
such that $f'''(c)\ge 6$ ", What I tried is that I draw graphs of $f(x) ,f'(x) , f''(x)$ and tried to use LMVT to conclude something useful for $f'''(x)$ ,but seems like graph way is too much messy and I know there are many cases which the graph can be for every type , so I thought some calculus way might be there leaving the graph technique.
 A: A third derivative is very hard to see from a graph.  It can make a sharp corner on a graph.  Your function starts out very flat at $0$ and ends up very flat at $2$.  If the third derivative is small the function can't rise very quickly from $0$ and won't have time to get to $2$.  You are told the Taylor series at $0$ is $kx^3$ plus higher order terms and the Taylor series at $2$ is $2+k'(x-2)^3$ plus higher order terms.
If there are no points where the third derivative is $6$ or greater, you can bound the second derivative by integrating $6$.  You can integrate twice more and bound the value of the function from above.  You can do the same starting from $2$ and going downwards.  If you can make the values and derivatives match you can exhibit a function that never touches $6$ and claim victory.
If not, you probably need $f'''(x) \ge 6$ over an interval, so the number of points is uncountable.  A more interesting question would be to show you can succeed with just one interval.  Intuitively you could jump with a(n almost) square wave from $0$ to almost $2$ with one bit of $f'''(x) \ge 6$.  Can you slow down enough to arrive at the other end without the third derivative getting large?
