Question

for all real valuea of x,the maximum value of the expression x/x^2-5x+9 is what?

Answer 1

Let x/(x²-5x+9) = y
=> yx² - (5y+1)x + 9y = 0.
For x to be real, discriminant ≥ 0
=> (5y+1)² - 36y² ≥ 0
=> 11y² - 10y - 1 ≤ 0
=> (11y + 1)(y-1) ≤ 0
=> -1/11 ≤ y ≤ 1

Remember: If (x-a)(x-b) ≤ 0, where b>a, then a ≤ x ≤ b.