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

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Question Number 34029 by rahul 19 last updated on 29/Apr/18
Number of integral values of x for which ((((π/2^(tan^(−1) x) )−4)(x−4)(x−10))/(x! − (x−1)!)) < 0
$$\boldsymbol{{N}}{umber}\:{of}\:{integral}\:{values}\:{of}\:{x}\:{for} \\ $$$${which}\: \\ $$$$\frac{\left(\frac{\pi}{\mathrm{2}^{\mathrm{tan}^{−\mathrm{1}} {x}} }−\mathrm{4}\right)\left({x}−\mathrm{4}\right)\left({x}−\mathrm{10}\right)}{{x}!\:−\:\left({x}−\mathrm{1}\right)!}\:<\:\mathrm{0} \\ $$
Commented by rahul 19 last updated on 29/Apr/18
x!= 1×2×3.........×(x−1)×x.
$${x}!=\:\mathrm{1}×\mathrm{2}×\mathrm{3}………×\left({x}−\mathrm{1}\right)×{x}. \\ $$
Answered by MJS last updated on 29/Apr/18
f(x)=((((π/2^(tan^(−1) x) )−4)(x−4)(x−10))/(x! − (x−1)!))<0 x!−(x−1)!>0 ∀ x∈N ⇒ ((π/2^(tan^(−1) x) )−4)(x−4)(x−10)<0 (π/2^(tan^(−1) x) )−4=((π−4×2^(tan^(−1) x) )/2^(tan^(−1) x) ) tan^(−1) x>0 ⇒ 2^(tan^(−1) x) >1 ⇒ ((π−4×2^(tan^(−1) x) )/2^(tan^(−1) x) )<0 ∀x∈N ⇒ (x−4)(x−10)>0 x−4<0 ∧ x−10<0 ⇒ x<4 x−4>0 ∧ x−10>0 ⇒ x>10 f(x)≥0 ⇒ 4≤x≤10 f(x) is not defined for x=0 ∧ x=1 f(x)<0 ⇒ x∈{2; 3}∪{x∈N∣x>10}
$${f}\left({x}\right)=\frac{\left(\frac{\pi}{\mathrm{2}^{\mathrm{tan}^{−\mathrm{1}} {x}} }−\mathrm{4}\right)\left({x}−\mathrm{4}\right)\left({x}−\mathrm{10}\right)}{{x}!\:−\:\left({x}−\mathrm{1}\right)!}<\mathrm{0} \\ $$$$ \\ $$$${x}!−\left({x}−\mathrm{1}\right)!>\mathrm{0}\:\forall\:{x}\in\mathbb{N} \\ $$$$\Rightarrow\:\left(\frac{\pi}{\mathrm{2}^{\mathrm{tan}^{−\mathrm{1}} {x}} }−\mathrm{4}\right)\left({x}−\mathrm{4}\right)\left({x}−\mathrm{10}\right)<\mathrm{0} \\ $$$$ \\ $$$$\frac{\pi}{\mathrm{2}^{\mathrm{tan}^{−\mathrm{1}} {x}} }−\mathrm{4}=\frac{\pi−\mathrm{4}×\mathrm{2}^{\mathrm{tan}^{−\mathrm{1}} \:{x}} }{\mathrm{2}^{\mathrm{tan}^{−\mathrm{1}} \:{x}} } \\ $$$$\mathrm{tan}^{−\mathrm{1}} \:{x}>\mathrm{0}\:\Rightarrow\:\mathrm{2}^{\mathrm{tan}^{−\mathrm{1}} \:{x}} >\mathrm{1}\:\Rightarrow\:\frac{\pi−\mathrm{4}×\mathrm{2}^{\mathrm{tan}^{−\mathrm{1}} \:{x}} }{\mathrm{2}^{\mathrm{tan}^{−\mathrm{1}} \:{x}} }<\mathrm{0}\:\forall{x}\in\mathbb{N} \\ $$$$\Rightarrow\:\left({x}−\mathrm{4}\right)\left({x}−\mathrm{10}\right)>\mathrm{0} \\ $$$$ \\ $$$${x}−\mathrm{4}<\mathrm{0}\:\wedge\:{x}−\mathrm{10}<\mathrm{0}\:\Rightarrow\:{x}<\mathrm{4} \\ $$$${x}−\mathrm{4}>\mathrm{0}\:\wedge\:{x}−\mathrm{10}>\mathrm{0}\:\Rightarrow\:{x}>\mathrm{10} \\ $$$$ \\ $$$${f}\left({x}\right)\geqslant\mathrm{0}\:\Rightarrow\:\mathrm{4}\leqslant{x}\leqslant\mathrm{10} \\ $$$${f}\left({x}\right)\:\mathrm{is}\:\mathrm{not}\:\mathrm{defined}\:\mathrm{for}\:{x}=\mathrm{0}\:\wedge\:{x}=\mathrm{1} \\ $$$${f}\left({x}\right)<\mathrm{0}\:\Rightarrow\:{x}\in\left\{\mathrm{2};\:\mathrm{3}\right\}\cup\left\{{x}\in\mathbb{N}\mid{x}>\mathrm{10}\right\} \\ $$
Commented by rahul 19 last updated on 29/Apr/18
You are getting this answer after assuming tan^(−1) x > 0 , right ? what if it′s <0 ?
$${You}\:{are}\:\mathrm{getting}\:\mathrm{this}\:\mathrm{answer}\:\mathrm{after}\: \\ $$$$\mathrm{assuming}\:\mathrm{tan}^{−\mathrm{1}} {x}\:>\:\mathrm{0}\:,\:{right}\:? \\ $$$${what}\:{if}\:{it}'{s}\:<\mathrm{0}\:? \\ $$
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
$$\mathrm{tan}^{−\mathrm{1}} \:{x}\geqslant\mathrm{0}\:\mathrm{for}\:\mathrm{all}\:{x}\geqslant\mathrm{0} \\ $$$$\mathrm{and}\:{x}!\:\mathrm{is}\:\mathrm{not}\:\mathrm{defined}\:\mathrm{for}\:{x}<\mathrm{0} \\ $$
Commented by rahul 19 last updated on 29/Apr/18
Ohh, yes! Thank you sir.
$${Ohh},\:{yes}! \\ $$$$\mathscr{T}{hank}\:{you}\:{sir}. \\ $$

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