<s>[INST] Calculus differentiable gra Hey guys so stuck on a calculus question. So far all I know is that $d$ should equal $4$. I then got the derivative of the a b c d function to $3ax^2 + 2bx + c$ , subbed in $(0,4)$ to get $c$, which was also $4$. Just wondering if I'm on the right track and where I should go from here.  

A section of road, represented by the line $y = x + 4$ when $x \leq 0$, is to
  be smoothly connected to another section of road, represented by $y = 4
> –x$ when $x \geq 4$ , by means of a curved section of road, represented by a
  cubic curve $y = ax^3+ bx^2+ cx + d$ . Find $a, b, c$ and $d$ such that the
  function $f(x)$ is everywhere differentiable (and therefore everywhere
  continuous), where
$$f(x) = \begin{cases}
x+4 &x\leq 0 \\ 
ax^3 +bx^2 +cx +d  &0<x<4 \\ 
 4-x &x\geq 4 
\end{cases}$$

 [/INST] you have $$ f(x) = \begin{cases}
x+4 &x\leq 0 \\ 
ax^3 +bx^2 +cx +d  &0<x<4 \\ 
 4-x &x\geq 4 
\end{cases}$$
making $f$ continuous at $x=0$ and at $x = 4$ requires $$d = 4,\quad 64a+16b+4c+d= 0\tag 1$$
we also have $$f'(x) = \begin{cases}
1 &x < 0 \\ 
3ax^2 +2bx +c & 0<x<4 \\ 
 -1 &x\geq 4 
\end{cases}$$
making $f'$ continuous at $x=0$ and at $x = 4$ requires $$c = 1,\quad 48a+8b+c= -1\tag 2$$
you can solve $(1)$ and $(2)$ to find all the constants $a, b, c, d.$
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