Maxima and MinimaMedium
Question
The maximum and minimum value of 
Options
A.2, 1
B.3, 1/3
C.1, 0
D.3, 1
Solution
Let y = 
x2y - xy + y = x2 + x + 1
x2(1 - y) + x(1 + y) + 1 - y = 0
for x ∈ R
D ≥ 0
(1 + y)2 - 4(1 - y) (1 - y) ≥ 0
1 + y2 + 2y - 4 (1 + y2 - 2y ) ≥> 0
y2 + 2y + 1 - 4y2 + 8y - 4 ≥ 0
-3y2 + 10y - 3 ≥ 0
3y2 - 10y + 3 ≤ 0
3y2 - 9y + y + 3 ≤ 0
3y (y - 3) - 1 (y - 3) ≤ 0
(3y - 1) (y - 3) ≤ 0
so max. = 3 min =
x2y - xy + y = x2 + x + 1
x2(1 - y) + x(1 + y) + 1 - y = 0
for x ∈ R
D ≥ 0
(1 + y)2 - 4(1 - y) (1 - y) ≥ 0
1 + y2 + 2y - 4 (1 + y2 - 2y ) ≥> 0
y2 + 2y + 1 - 4y2 + 8y - 4 ≥ 0
-3y2 + 10y - 3 ≥ 0
3y2 - 10y + 3 ≤ 0
3y2 - 9y + y + 3 ≤ 0
3y (y - 3) - 1 (y - 3) ≤ 0
(3y - 1) (y - 3) ≤ 0
so max. = 3 min =
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