Number-of-integral-values-of-x-for-which-pi-2-tan-1-x-4-x-4-x-10-x-x-1-lt-0-

Question Number 34029 by rahul 19 last updated on 29/Apr/18

N u m b e r o f i n t e g r a l v a l u e s o f x f o r

w h i c h

\frac{(\frac{π}{2^{\tan^{- 1} x}} - 4) (x - 4) (x - 10)}{x! - (x - 1)!} < 0

Commented by rahul 19 last updated on 29/Apr/18

x! = 1 \times 2 \times 3 \dots \dots \dots \times (x - 1) \times x .

Answered by MJS last updated on 29/Apr/18

f (x) = \frac{(\frac{π}{2^{\tan^{- 1} x}} - 4) (x - 4) (x - 10)}{x! - (x - 1)!} < 0

x! - (x - 1)! > 0 \forall x \in N

\Rightarrow (\frac{π}{2^{\tan^{- 1} x}} - 4) (x - 4) (x - 10) < 0

\frac{π}{2^{\tan^{- 1} x}} - 4 = \frac{π - 4 \times 2^{\tan^{- 1} x}}{2^{\tan^{- 1} x}}

\tan^{- 1} x > 0 \Rightarrow 2^{\tan^{- 1} x} > 1 \Rightarrow \frac{π - 4 \times 2^{\tan^{- 1} x}}{2^{\tan^{- 1} x}} < 0 \forall x \in N

\Rightarrow (x - 4) (x - 10) > 0

x - 4 < 0 \land x - 10 < 0 \Rightarrow x < 4

x - 4 > 0 \land x - 10 > 0 \Rightarrow x > 10

f (x) ⩾ 0 \Rightarrow 4 ⩽ x ⩽ 10

f (x) is not defined for x = 0 \land x = 1

f (x) < 0 \Rightarrow x \in {2; 3} \cup {x \in N ∣ x > 10}

Commented by rahul 19 last updated on 29/Apr/18

Y o u a r e getting this answer after

assuming \tan^{- 1} x > 0, r i g h t ?

w h a t i f {i t}^{'} s < 0 ?

Commented by MJS last updated on 29/Apr/18

\tan^{- 1} x ⩾ 0 for all x ⩾ 0

and x! is not defined for x < 0

Commented by rahul 19 last updated on 29/Apr/18

O h h, y e s!

T h a n k y o u s i r .