f(1) = a + b + c = -a/2 => 3a + 2b + 2c = 0
if 3a < 0, then 0 > 3a > 2b > 2C => 3a + 2b + 2c < 0, sothis condition is false;
if 3a = 0, then 0= 3a > 2c > 2b => 2c + 2b < 0, this condition is also false.
hence 3a > 0 => a > 0;
OR we have 3a + 2b + 2c = 0 < 9a => a > 0;
then because c > b, 0= 3a + 2b + 2c > 3a + 2b + 2b => 3a + 4b < 0;
Similarly, 3a > 2c => 0= 3a + 2b + 2c < 6a + 2b => b > -3a
(2). Because a > 0, then f(1) = -a/2 < 0, then f(2) = 4a + 2b + c = 3a + 2b + 2c + a - c = a - c
if c < 0 => a - c > 0 ,
if c = 0 => a - c = a > 0,
if c > 0 => f(0) = c > 0.
So no matter what value c takes, f(0)f(1) < 0 OR f(2)f(1) < 0, => at least we have one root in (0,2).
The End