If-f-x-2-x-a-x-1-is-monotone-function-find-the-range-of-a-

Question Number 205490 by mnjuly1970 last updated on 22/Mar/24

I f, f (x) = \sqrt{2 + x} + a \sqrt{x - 1}

i s m o n o t o n e f u n c t i o n .

f i n d t h e r a n g e o f^{″} a^{″}

Answered by mr W last updated on 22/Mar/24

x ⩾ 1

f (x) = \sqrt{3 + x - 1} + a \sqrt{x - 1}

l e t t = x - 1 ⩾ 0

f (t) = \sqrt{3 + t^{2}} + a t

f^{'} (t) = \frac{t}{\sqrt{3 + t^{2}}} + a = \frac{1}{\sqrt{1 + \frac{3}{t^{2}}}} + a

s u c h t h a t t h e f u n c t i o n i s m o n o t o n e

f o r t ⩾ 0, \frac{1}{\sqrt{1 + \frac{3}{t^{2}}}} + a = 0 s h o u l d h a v e

n o s o l u t i o n . t h i s i s t h e c a s e :

1) i f a ⩾ 0

2) i f a ⩽ - 1

i . e . t h e r a n g e o f a i s a ⩾ 0 o r a ⩽ - 1

o r

f u n c t i o n i s n o t m o n o t o n e i f

- 1 < a < 0

Commented by mnjuly1970 last updated on 22/Mar/24

t h a n k s a l o t s i r \dots