Question

if x is real prove that x/x'2-5x+9​ lies on -1/11 & 1

Answer 1

Answer:

let

x

2

−5x+9

x

=y

x=yx

2

−(5y)x+9y

yx

2

−(1+5y)x+9y

For this quadratic equation to have real roots

Discriminant must be ≥0

(−1−5y)

2

−4×y×9y≥0

1+25y

2

+10y−36y

2

≥0

1+10y−11y

2

≥0

1−y+11y−11y

2

≥0

(1−y)+11y(1−y)≥0

(11y+1)(y−1)≤0

So (y−(

11

−1

))(y−1)≤0

So y∈[ 11−1 ,1]