If-x-R-the-maximum-value-of-3x-2-9x-17-3x-2-9x-7-is-

Question Number 162371 by cortano last updated on 29/Dec/21

I f x \in R t h e m a x i m u m v a l u e

o f \frac{3 x^{2} + 9 x + 17}{3 x^{2} + 9 x + 7} i s \dots

Answered by mindispower last updated on 29/Dec/21

g (x) = \frac{3 x^{2} + 9 x + 17}{3 x^{2} + 9 x + 7} = 1 + \frac{10}{3 x^{2} + 9 x + 7} = 1 + \frac{10}{3 {(x + \frac{3}{2})}^{2} + \frac{1}{4}}

⩽ 1 + \frac{10}{\frac{1}{4}} = 41 (x = - \frac{3}{2})

Answered by bobhans last updated on 29/Dec/21

y = \frac{{3 x}^{2} + 9 x + 17}{{3 x}^{2} + 9 x + 7} \Rightarrow {3 yx}^{2} + 9 yx + 7 y = {3 x}^{2} + 9 x + 17

\Rightarrow 3 (y - 1) x^{2} + 9 (y - 1) x + 7 y - 17 = 0

\Rightarrow Δ ⩾ 0

\Rightarrow 81 {(y - 1)}^{2} - 12 (y - 1) (7 y - 17) ⩾ 0

\Rightarrow (y - 1) (27 y - 27 - 28 y + 68) ⩾ 0

\Rightarrow (y - 1) (- y + 41) ⩾ 0

\Rightarrow (y - 1) (y - 41) ⩽ 0

\Rightarrow 1 ⩽ y ⩽ 41 \to {\begin{cases} \max = 41 \\ \min = 1 \end{cases}

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