<s>[INST] If the range of a function is $[a,b]$ ,where $a,b \in \mathbb N$ ,find $f(a+b)$. Let  $f(x)= (x+1) (x+2) (x+3) (x+4) + 5$  where $x \in [-6,6]$. If the range of this function is $[a,b]$ ,where $a,b \in \mathbb  N$ ,find $f(a+b)$.
ATTEMPT:
Sketching the graph the we know that minima of this function is between $-1$ and $-2$, and since the function is strictly increasing in the interval $[-1,6]$, maxima will be at $6$.
But how to get the minima?
 [/INST] Hint
$$(x+1)(x+2)(x+3)(x+4)+5=[(x+1)(x+4)][(x+2)(x+3)]+5=(x^2+5x+4)(x^2+5x+6)+5
=(x^2+5x+5)^2+4$$
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