y-ax-3-bx-2-cx-d-then-prove-that-the-equation-y-0-has-only-one-real-root-if-a-9ad-bc-2-4-b-2-3ac-c-2-3bd-gt-0-provided-b-2-gt-3ac-

Question Number 24565 by ajfour last updated on 21/Nov/17

y = {a x}^{3} + {b x}^{2} + c x + d, t h e n

p r o v e t h a t t h e e q u a t i o n y = 0

h a s o n l y o n e r e a l r o o t i f

a [{(9 a d - b c)}^{2} - 4 (b^{2} - 3 a c) (c^{2} - 3 b d)]

> 0 p r o v i d e d b^{2} > 3 a c .

Answered by ajfour last updated on 21/Nov/17

Commented by ajfour last updated on 21/Nov/17

I f t h e l o c a l m i n i m u m v a l u e

a n d t h e l o c a l m a x i m u m v a l u e,

b o t h, a r e o f t h e s a m e s i g n, t h e n,

i b e l i e v e, t h e r e c a n b e j u s t o n e

r e a l r o o t o f a c u b i c e q u a t i o n .

y = {a x}^{3} + {b x}^{2} + c x + d

\Rightarrow \frac{d y}{d x} = 3 {a x}^{2} + 2 b x + c

l e t a t x = α, β \frac{d y}{d x} = 0

\Rightarrow α β = \frac{c}{3 a} a n d (α + β) = - \frac{2 b}{3 a}

\Rightarrow 3 a α^{2} + 2 b α + c = 0 \dots . . (i)

3 a β^{2} + 2 b β + c = 0 \dots . . (i i)

F o r o n e r e a l r o o t

y (α) \times y (β) > 0

o r 3 y (α) \times 3 y (β) > 0

3 y (α) = 3 a α^{3} + 3 b α^{2} + 3 c α + 3 d

s u b t r a c t i n g α \times (i) f r o m t h i s

3 y (α) = b α^{2} + 2 c α + 3 d

= \frac{b}{3 a} (3 a α^{2}) + 2 c α + 3 d

u s i n g (i) a g a i n :

3 y (α) = - \frac{b}{3 a} (2 b α + c) + 2 c α + 3 d

= 2 α (c - \frac{b^{2}}{3 a}) + (3 d - \frac{b c}{3 a})

s o 3 y (α) \times 3 y (β) =

[4 α β {(c - \frac{b^{2}}{3 a})}^{2} + 2 (α + β) (c - \frac{b^{2}}{3 a}) (3 d - \frac{b c}{3 a})

+ {(3 d - \frac{b c}{3 a})}^{2}]

A s α = \frac{c}{3 a} a n d β = - \frac{2 b}{3 a} w e h a v e

3 y (α) \times 3 y (β) =

\frac{4 c}{3 a} {(c - \frac{b^{2}}{3 a})}^{2} - \frac{4 b}{3 a} (c - \frac{b^{2}}{3 a}) (3 d - \frac{b c}{3 a})

+ {(3 d - \frac{b c}{3 a})}^{2} > 0

o r

4 c {(3 a c - b^{2})}^{2} - 4 b (3 a c - b^{2}) (9 a d - b c)

+ 3 a {(9 a d - b c)}^{2} > 0

o r

3 a {(9 a d - b c)}^{2} + 4 (3 a c - b^{2}) (3 {a c}^{2} - b^{2} c

- 9 a b d + b^{2} c) > 0

\Rightarrow

3 a {(9 a d - b c)}^{2} + 4 (3 a) (3 a c - b^{2}) (c^{2} - 3 b d) > 0

a [{(9 a d - b c)}^{2} - 4 (b^{2} - 3 a c) (c^{2} - 3 b d)] > 0 .

Facebook Tweet Pin