Solve-for-real-numbees-a-16-1-a-b-2-b-16-1-b-c-2-c-16-1-c-x-2-

Question Number 138783 by mathdanisur last updated on 18/Apr/21

S o l v e f o r r e a l n u m b e e s :

{\begin{cases} a^{16} + 1 = \frac{a + b}{2} \\ b^{16} + 1 = \frac{b + c}{2} \\ c^{16} + 1 = \frac{c + x}{2} \end{cases}

Commented by mr W last updated on 18/Apr/21

n o r e a l r o o t s!

Commented by mathdanisur last updated on 18/Apr/21

D e a r S i r, c a n y o u t e l l m e w h y t h e r e i s n o r e a l r o o t ?

Commented by mr W last updated on 18/Apr/21

i f a = t i s a r o o t, t h e n b = t a n d c = t a r e

a l s o r o o t s .

t^{16} + 1 = \frac{t + t}{2}

t^{16} - t + 1 = 0 ?

f (t) = t^{16} - t + 1

f^{'} (t) = 16 t^{15} - 1 = 0 \Rightarrow t = \frac{1}{\sqrt[15]{16}}

f^{″} (t) = 16 \times 15 t^{14} > 0

i . e . f_{m i n} i s f (\frac{1}{\sqrt[15]{16}}) = \frac{1}{16^{\frac{16}{15}}} - \frac{1}{16^{\frac{1}{15}}} + 1 \approx 0.2207

s i n c e t^{16} - t + 1 ⩾ 0.2207

t h e r e f o r e t^{16} - t + 1 = 0 h a s n o r e a l r o o t s .

Commented by mathdanisur last updated on 18/Apr/21

T h a n k y o u v e r y m u c h d e a r S i r