The-set-of-values-of-a-for-which-all-the-solutions-of-the-equation-4sin-4-x-asin-2-x-3-0-are-real-and-distinct-is-

Question Number 32563 by rahul 19 last updated on 27/Mar/18

T h e s e t o f v a l u e s o f^{'} a^{'} f o r w h i c h

a l l t h e s o l u t i o n s o f t h e e q u a t i o n

4 \sin^{4} x + a \sin^{2} x + 3 = 0 a r e r e a l a n d

d i s t i n c t i s ?

Commented by rahul 19 last updated on 27/Mar/18

i t r i e d d o i n g t h e s a m e p r o c e d u r e

a s i n Q u e s . N o . 32220 .

Commented by rahul 19 last updated on 27/Mar/18

C o r r e c t a n s . i s [- 7, - 4 \sqrt{3}) .

Answered by rahul 19 last updated on 28/Mar/18

D ⩾ 0

a^{2} ⩾ 48

\Rightarrow a ⩾ 4 \sqrt{3} a n d a ⩽ - 4 \sqrt{3}

N o w l e t {s i n}^{2} x = t

\Rightarrow 4 t^{2} + a t + 3 = 0

\Rightarrow f (0) . f (1) < 0

\Rightarrow 3 \times (7 + a) < 0

\Rightarrow a < - 7 .

W {h a t}^{'} s w r o n g w i t h m y s o l u t i o n ?

Commented by rahul 19 last updated on 29/Mar/18

w h a t d o e s \lor, \land m e a n s i n y o u r

s o l u t i o n ?

Commented by MJS last updated on 29/Mar/18

I found it was off topic

look below

Commented by MJS last updated on 29/Mar/18

\land = “ {a n d}^{″}

\lor = “ {o r}^{″}

Answered by MJS last updated on 28/Mar/18

s^{2} + \frac{a}{4} s + \frac{3}{4} = 0

exactly 1 solution :

{(s - s_{0})}^{2} = s^{2} - 2 s_{0} s + s_{0}^{2}

- 2 s_{0} = \frac{a}{4} \land s_{0}^{2} = \frac{3}{4}

(a = 4 \sqrt{3} \land s = - \frac{\sqrt{3}}{2}) \lor (a = - 4 \sqrt{3} \land s = \frac{\sqrt{3}}{2})

BUT :

t^{4} + \frac{a}{4} t^{2} + \frac{3}{4} = 0 \Rightarrow t = \sqrt{s} \Rightarrow s ⩾ 0

\Rightarrow a = - 4 \sqrt{3} \land s = \frac{\sqrt{3}}{2} \land t = \pm \frac{\sqrt[4]{3} \sqrt{2}}{2}

this shows that the equation

f (x) = {4 \sin}^{4} x - 4 \sqrt{3} \sin^{2} x + 3 has

exactly 1 solution in [0; \frac{π}{2}]

with a > - 4 \sqrt{3} it has none

with - 7 ⩽ a < - 4 \sqrt{3} it has 2

with a < - 7 it has 1 again

a \to - \infty \Rightarrow x_{0} \to 0 but f (0) = 3 \forall a \in R

Commented by rahul 19 last updated on 28/Mar/18

s i r p l s c h e c k m y m e t h o d .

Commented by MJS last updated on 29/Mar/18

your method is ok but the

conclusions might be wrong

in qu . 32220 we were looking

for solutions, here {we}^{'} re looking

for all distinct solutions, so there

must be 4 in [0; π]

it seems difficult to understand

the nature of the connection

between the functions

f (x) = c_{1} \sin^{4} x + c_{2} \sin^{2} x + c_{3}

g (s) = c_{1} s^{4} + c_{2} s^{2} + c_{3}

h (t) = c_{1} t^{2} + c_{2} t + c_{3}

you should draw them or plot

them, if possible .

Commented by MJS last updated on 29/Mar/18

Commented by MJS last updated on 29/Mar/18

Commented by MJS last updated on 29/Mar/18

Commented by MJS last updated on 29/Mar/18

{we}^{'} re losing important information :

1 . picture

{4 \sin}^{4} x - 4 \sqrt{3} \sin^{2} x + 3

{4 \sin}^{4} x - {7 \sin}^{2} x + 3

[\frac{π}{4}; \frac{3 π}{4}]

2 . picture

4 s^{4} - 4 \sqrt{3} s^{2} + 3

4 s^{4} - 7 s^{2} + 3

[\frac{\sqrt{2}}{2}; 1 . 11^{(*)}]

3 . picture

4 t^{2} - 4 \sqrt{3} t + 3

4 t^{2} - 7 t + 3

[\frac{\sqrt{2}}{2}; 1 . 02^{(*)}]

(*) upper borders chosen for

symmetry of pics

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