Question

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

• A. $5049$
• B. $5045$
• C. $5041$
• D. None of these

Answer 1

Answer: (A)

$5049$

$f\left(x\right)=(x+1)(x+2)(x+3)(x+4)+5$

$x\in [-6,6]$

$=(x^2+5x+4)(x^2+5x+6)+5$

Let $t=x^2+5x$

$(t+4)(t+6)+5=t^2+10t+24+5$

$=t^2+10t+29$

$=(t+5{)}^2+4$

$f\left(x\right)=(x^2+5x+5{)}^2+4$

$4\le f\left(x\right)\le (36+30+5{)}^2+4$

$4\le f\left(x\right)\le 5041+4=5045$

$a+b=5049$
Ans: A