if-f-x-2x-2-12x-10-i-sketch-the-graph-of-y-f-x-for-1-x-7-ii-find-the-set-of-values-of-k-for-which-the-equation-f-x-k-has-4-distinct-roots-

Question Number 113854 by ayenisamuel last updated on 15/Sep/20

i f f (x) = 2 x^{2} - 12 x + 10 .

(i) s k e t c h t h e g r a p h o f y =∣ f (x) ∣ f o r

- 1 ⩽ x ⩽ 7 .

(i i) f i n d t h e s e t o f v a l u e s o f k f o r

w h i c h t h e e q u a t i o n ∣ f (x) ∣= k h a s 4

d i s t i n c t r o o t s .

Answered by MJS_new last updated on 15/Sep/20

∣ 2 x^{2} - 12 x + 10 ∣= k \Rightarrow k ⩾ 0

{(2 x^{2} - 12 x + 10)}^{2} = k^{2}

x^{4} - 12 x^{3} + 46 x^{2} - 60 x - \frac{k^{2}}{4} + 25 = 0

let x = t + 3

t^{4} - 8 t^{2} - \frac{k^{2}}{4} + 16 = 0

\Rightarrow t^{2} = \frac{8 \pm k}{2} \Rightarrow t = \pm \sqrt{\frac{8 \pm k}{2}} [4 solutions]

\Rightarrow x_{1, 2} = 3 \pm \sqrt{\frac{8 - k}{2}}; x_{3, 4} = 3 \pm \sqrt{\frac{8 + k}{2}}

\Rightarrow 4 solutions only if 8 - k > 0 \land 8 + k > 0

\Rightarrow - 8 < k < 8 but k ⩾ 0

\Rightarrow 0 ⩽ k < 8

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