Question

Let $f( x )=( x +1)( x +2)( x +3)( x +4)+5$ where $x \in[-6,6]$. If the range of the function is $[a, b]$ where $a, b \in N$ then find the value of $(a+b)$.

Answer 1

$ f ( x )=\left( x ^2+5 x +4\right)\left( x ^2+5 x +6\right)+5$
$=\left[\left(x^2+5 x+5\right)-1\right]\left[\left(x^2+5 x+5\right)+1\right]+5$
$=\left(x^2+5 x+5\right)^2-1+5$
$f(x)=\left(x^2+5 x+5\right)^2+4$
Hence $f(x)$ has a minimum value 4 when $x^2+5 x+5=0$
i.e. $ x=\frac{-5 \pm \sqrt{5}}{2} ;$ Hence $x=\frac{-(5+\sqrt{5})}{2} \in[-6,6]$ also maximum occurs at $x =6$
$\left. f ( x )\right|_{\max }=(36+30+5)^2+4=(71)^2+4=5041+4=5045$
range is $[4,5045]$
$\therefore a =4 ; b =5045 \Rightarrow a + b =5049 $
Alternatively: $f ( x )-5= g ( x )]$