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

Question Number 27524 by julli deswal last updated on 08/Jan/18

If x is real, the maximum value of

\frac{3 x^{2} + 9 x + 17}{3 x^{2} + 9 x + 7} is

Answered by prakash jain last updated on 08/Jan/18

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

3 x^{2} y + 9 x y + 7 y = 3 x^{2} + 9 x + 17

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

x \in R \Rightarrow Δ ⩾ 0

81 {(y - 1)}^{2} - 4 \times 3 (y - 1) (7 y - 17) ⩾ 0

81 (y^{2} - 2 y + 1) - 12 (7 y^{2} - 24 y + 17) ⩾ 0

- 3 y^{2} + 126 y - 123 ⩾ 0

- y^{2} + 42 y - 41 ⩾ 0

- (y - 41) (y - 1) ⩾ 0

1 ⩽ y ⩽ 41

\min . value of y = 1

\max . value of y = 41