Find-local-minimum-of-function-H-x-34-3-2x-x-2-3x-1-

Question Number 148676 by bemath last updated on 30/Jul/21

Find local minimum of function

H (x) = \frac{34}{3 + \frac{2 x}{x^{2} + 3 x + 1}} .

Answered by EDWIN88 last updated on 30/Jul/21

H (x) = \frac{34 x^{2} + 102 x + 34}{3 x^{2} + 11 x + 3}

3 x^{2} h + 11 h x + 3 h = 34 x^{2} + 102 x + 34

(34 - 3 h) x^{2} + (102 - 11 h) x + 34 - 3 h = 0

Δ ⩾ 0

{(102 - 11 h)}^{2} - 4 (34 - 3 h) (34 - 3 h) ⩾ 0

10404 - 2244 h + 121 h^{2} - 4 (1156 - 204 h + 9 h^{2}) ⩾ 0

121 h^{2} - 2244 h + 10404 - 4624 + 816 h - 36 h^{2} ⩾ 0

85 h^{2} - 1428 h + 5780 ⩾ 0

(h - 10) (5 h - 34) ⩾ 0

h ⩽ \frac{34}{5} \cup h ⩾ 10

H {(x)}_{m i n} = 10 w h e n x = 1

Commented by EDWIN88 last updated on 30/Jul/21

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