Find-valu-of-x-if-x-R-9x-1-1-3-8x-1-8x-15-1-4-5-2-0-

Question Number 162043 by HongKing last updated on 25/Dec/21

Find valu of x if x \in R

\sqrt[3]{9 x - 1} + \sqrt{8 x - 1} + \sqrt[4]{8 x + 15} - \frac{5}{2} = 0

Commented by mr W last updated on 25/Dec/21

w i t h x = \frac{1}{8}

\sqrt[3]{\frac{9}{8} - 1} + \sqrt{\frac{8}{8} - 1} + \sqrt[4]{\frac{8}{8} + 15} - \frac{5}{2}

= \sqrt[3]{\frac{1}{8}} + \sqrt{0} + \sqrt[4]{16} - \frac{5}{2}

= \frac{1}{2} + 0 + 2 - \frac{5}{2}

= 0 ✓

\Rightarrow x = \frac{1}{8} i s o n e a n d t h e o n l y o n e r o o t,

s i n c e f (x) i s s t r i c t l y i n c r e a s i n g .

Commented by HongKing last updated on 26/Dec/21

thank you my dear Sir cool

Answered by aleks041103 last updated on 25/Dec/21

f (x) = \sqrt[3]{9 x - 1} + \sqrt{8 x - 1} + \sqrt[4]{8 x + 15}

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

\Rightarrow t h e r e i s a t m o s t 1 s o l u t i o n t o f (x) = \frac{5}{2}

f (x) = \frac{5}{2}

\Rightarrow \sqrt[3]{72 x - 8} + \sqrt{32 x - 4} + \sqrt[4]{128 x + 240} = 5

f o r f (x) \in R \Rightarrow

32 x - 4 ⩾ 0 \Rightarrow x ⩾ \frac{1}{8}

128 x + 240 ⩾ 0 \Rightarrow x ⩾ - \frac{240}{128}

\Rightarrow x ⩾ \frac{1}{8}

t r y x = \frac{1}{8}

\Rightarrow \sqrt[3]{72 x - 8} + \sqrt{32 x - 4} + \sqrt[4]{128 x + 240} =

= \sqrt[3]{9 - 8} + \sqrt{4 - 4} + \sqrt[4]{16 + 240} =

= 1 + 0 + 4 = 5

\Rightarrow x = \frac{1}{8} i s a s o l u t i o n

\Rightarrow t h e o n l y s o l u t i o n o f f (x) = \frac{5}{2} o v e r R

i s x = \frac{1}{8}

Commented by HongKing last updated on 25/Dec/21

cool my dear Sir thank you so much