solve-W-In-4x-x-1-

Question Number 183712 by Michaelfaraday last updated on 29/Dec/22

s o l v e :

W (I n (4 x)) = \sqrt{(x - 1)}

Answered by mr W last updated on 29/Dec/22

e^{\ln (4 x)} \ln (4 x) = \sqrt{x - 1}

{(4 x)}^{4 x} = e^{\sqrt{x - 1}}

l e t t = \sqrt{x - 1} ⩾ 0

{(4 t^{2} + 4)}^{(4 t^{2} + 4)} = e^{t}

f (t) = {(4 t^{2} + 4)}^{(4 t^{2} + 4)}

g (t) = e^{t}

f (0) - g (0) = 4^{4} - e^{0} = 256 - 1 = 255

f^{'} (t) - g^{'} (t) = \dots > 0

\Rightarrow f (t) - g (t) i s s t r i c t l y i n c r e a s i n g,

i . e . f (t) - g (t) ⩾ 255 f o r t ⩾ 0

\Rightarrow f (t) - g (t) = 0 h a s n o r o o t s!

\Rightarrow W (I n (4 x)) = \sqrt{x - 1} h a s n o r o o t s!

Commented by Michaelfaraday last updated on 29/Dec/22

w o w, t h a n k s s i r f o r y o u r h e l p