FunctionHard
Question
For real x, the function 
will assume all real values provided.
will assume all real values provided. Options
A.a > b > c
B.a < b < c
C.a > c < b
D.a ≤ c ≤ b
Solution
Let 
⇒ yx - cy = x2 - (a + b)x + ab
⇒ x2 - (a + b + y)x + (ab + cy) = 0
For real roots, D ≥ 0
⇒ (a + b + y)2 - 2(ab + cy) ≥ 0
⇒ (a + b)2 + y + 2(a + b) y - 4ab - 4cy ≥ 0
⇒ y2 + 2(a + b - 2c)y + (a - b)2 ≥ 0
Which is true for all real values of y,
∴ D ≤ 0
⇒ 4(a + b - 2c)2 - 4(a - b)2 ≤ 0
⇒ 4(a + b - 2c + a - b)(a + b - 2c - a + b) ≤ 0
⇒ (2a - 2c)(2b - 2c) ≤ 0
⇒ (a - c)(b - c) ≤ 0
⇒ (c - a)(c - b) ≤ 0
⇒ c must lie between a and b
ie, a ≤ c ≤ b or b ≤ c ≤ a
⇒ yx - cy = x2 - (a + b)x + ab
⇒ x2 - (a + b + y)x + (ab + cy) = 0
For real roots, D ≥ 0
⇒ (a + b + y)2 - 2(ab + cy) ≥ 0
⇒ (a + b)2 + y + 2(a + b) y - 4ab - 4cy ≥ 0
⇒ y2 + 2(a + b - 2c)y + (a - b)2 ≥ 0
Which is true for all real values of y,
∴ D ≤ 0
⇒ 4(a + b - 2c)2 - 4(a - b)2 ≤ 0
⇒ 4(a + b - 2c + a - b)(a + b - 2c - a + b) ≤ 0
⇒ (2a - 2c)(2b - 2c) ≤ 0
⇒ (a - c)(b - c) ≤ 0
⇒ (c - a)(c - b) ≤ 0
⇒ c must lie between a and b
ie, a ≤ c ≤ b or b ≤ c ≤ a
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