Solve-the-equation-2-x-2-x-4-1-2-x-3-x-4-1-x-4-1-

Question Number 190928 by TUN last updated on 14/Apr/23

S o l v e t h e e q u a t i o n :

2 + x^{2} (x^{4} + 1) = \frac{\sqrt{2} x^{3} (x^{4} - 1)}{\sqrt{x^{4} + 1}}

Commented by Frix last updated on 14/Apr/23

x = \pm i

Commented by TUN last updated on 14/Apr/23

i t m e a n s n o r e a l r o o t

Commented by TUN last updated on 14/Apr/23

p r o v e i t

Commented by Frix last updated on 14/Apr/23

2 + x^{2} (x^{4} + 1) - \frac{\sqrt{2} x^{3} (x^{4} - 1)}{\sqrt{x^{4} + 1}} ⩾ 2

Commented by TUN last updated on 15/Apr/23

h o w t o ⩾ 2

Commented by Frix last updated on 15/Apr/23

2 + x^{2} (x^{4} + 1) - \frac{\sqrt{2} x^{3} (x^{4} - 1)}{\sqrt{x^{4} + 1}} ⩾ 2

x^{2} (x^{4} + 1) - \frac{\sqrt{2} x^{3} (x^{4} - 1)}{\sqrt{x^{4} + 1}} ⩾ 0

{(x^{4} + 1)}^{\frac{3}{2}} - \sqrt{2} x (x^{4} - 1) ⩾ 0

f (x) = {(x^{4} + 1)}^{\frac{3}{2}} - \sqrt{2} x (x^{4} - 1)

f^{'} (x) = 0 \Rightarrow x \approx - . 522499270

f^{″} (- . 522499270) > 0 \Rightarrow

Minimum (f (x)) = . 430 \dots > 0

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