Christmas Streak 60/88: 2018 CSAT (Korean SAT) #29 of Liberal Arts

For two reals $a$ and $k,$ two functions $f(x)$ and $g(x)$ are defined as:

$\begin{aligned} f(x) &=\begin{cases} 0, & x\le a \\ (x-1)^2(2x+1), & x>a\end{cases}\\ g (x)& =\begin{cases} 0, & x\le k \\ 12(x-k), & x>k \end{cases} \end{aligned}$

Given the two conditions below, the minimum of $k$ is $\dfrac{p}{q},$ for some coprime positive integers $p$ and $q.$

Find the value of $a+p+q.$

(1) Function $f(x)$ is differentiable over all reals.

(2) For all $x,$ $f(x)\ge g(x).$

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This problem is a part of <Christmas Streak 2017> series .