Continuity and derivability of a piece wise f unction Let $f(x)=x^3-9x^2+15x+6$ and 
$$g(x)=
\begin{cases} 
      \min f(t) &\mbox{if } 0\leq t \leq x, 0 \leq x \leq6 \\
      x-18 &\mbox{if }x\geq 6. \\ 
\end{cases}
$$
Then discuss the continuity and derivability of $g(x)$. 
Could someone explain be how to deal with $\min f(t)$ where  $0\leq t \leq x, 0 \leq x \leq6$.

To deal with the minimum, I think you should study the variations (by using the derivation sounds fine) of f in [0, 6]. As you probably noticed, f is a polynomial and the minimum in the definition of is written with large inequalities, which means that the minimum will be reached (a polynomial being continuous, and a continuous function being uniforlmy continuous on a closed interval).
Like so, you will be able to write g with either a constant value or f itself on [0, 6] (and probably both if the minimum in not reached in 0 or 6).
P.S : there shouldn't be "=0" at the end of the definition of f, right ?
