Find-the-number-of-real-roots-of-the-equation-f-x-x-3-2x-2-2x-1-0-

Question Number 20259 by Tinkutara last updated on 24/Aug/17

Find the number of real roots of the

equation f (x) = x^{3} + 2 x^{2} + 2 x + 1 = 0

Answered by ajfour last updated on 24/Aug/17

f^{'} (x) = 3 x^{2} + 4 x + 2 > 0

s o j u s t o n e r e a l r o o t .

Commented by Tinkutara last updated on 25/Aug/17

Thank you very much Sir!

Commented by Tinkutara last updated on 24/Aug/17

Why f^{'} (x) > 0 \Rightarrow one real root only ?

Commented by ajfour last updated on 24/Aug/17

Commented by ajfour last updated on 24/Aug/17

i n o r d e r t o c r o s s x - a x i s a g a i n t h e

c u r v e n e e d s t o t a k e a t u r n, t h e

p o i n t w h e r e t a n g e n t i s h o r i z o n t a l

f^{'} (x) = \frac{d y}{d x} = 0, i f p r i o r t o t h i s p o i n t

f^{'} (x) < 0 a n d a f t e r t h a t f^{'} (x) > 0

o r f i r s t f^{'} (x) > 0 t h e n t h r o u g h

z e r o i t b e c o m e s < 0 t h e n i t

i m p l i e s a t u r n i s t a k e n . l i n e t a k e s

n o t u r n h a s m a x i m u m o n e r o o t

(e x c e p t y = 0), a q u a d r a t i c

f u n c t i o n t a k e s o n e t u r n, m a x

t w o r o o t s,

c u b i c f u n c t i o n p r o c e e d s f r o m

\mp \infty t o \pm \infty . i n b e t w e e n i f i t

d o e s n o t t a k e t h i s t u r n i t c r o s s e s

x o n l y o n c e . A n d s o m e t i m e s

t h e r e i s t h e t u r n b u t y e t i t c r o s s e s

x - a x i s o n l y o n c e, a s (i n s k e t c h i n

c o m m e n t) . A n y h o w i f f (x) i s

c u b i c a n d f^{'} (x) n e v e r z e r o t h e n i t

e i t h e r k e e p s i n c r e a s i n g (i f c o e f f .

o f x^{3} + v e) o r k e e p s o n l y

d e c r e a s i n g (i f c o e f f . o f x^{3} - v e) . .

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