What-is-range-of-function-f-x-x-1-x-2-x-3-x-4-1-where-x-1-1-

Question Number 136270 by EDWIN88 last updated on 20/Mar/21

What is range of function f (x) = (x + 1) (x + 2) (x + 3) (x + 4) + 1

where x \in [- 1, 1]

Answered by mr W last updated on 20/Mar/21

f (x) = (x^{2} + 5 x + 4) (x^{2} + 5 x + 6) + 1

f (x) = (x^{2} + 5 x + 4) (x^{2} + 5 x + 4 + 2) + 1

f (x) = {(x^{2} + 5 x + 4)}^{2} + 2 (x^{2} + 5 x + 4) + 1

f (x) = {(x^{2} + 5 x + 5)}^{2}

f (x) = {[{(x + \frac{5}{2})}^{2} - \frac{5}{4}]}^{2} ⩾ 0

{(x + \frac{5}{2})}^{2} - \frac{5}{4} = 0

x = - \frac{5 + \sqrt{5}}{2} \approx - 3.618, - \frac{5 - \sqrt{5}}{2} \approx - 1.382

i . e . i n (- \infty, - 3.618] f (x) i s d e c r e a s i n g

i n [- 1.382, + \infty) f (x) i s i n c r e a s i n g

\Rightarrow i n [- 1, 1] :

m i n . f (x) = f (- 1) = 1

m a x . f (x) = f (1) = 2 \times 3 \times 4 \times 5 + 1 = 121

i . e . r a n g e o f f (x) i n [- 1, 1] i s [1, 121] .

Commented by mr W last updated on 20/Mar/21

Commented by EDWIN88 last updated on 20/Mar/21

great . .

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