If `2a+3b+6c=0,` then prove that at least one root of the equation `a x^2+b x+c=0` lies in the interval (0,1).

Explanation

We have to prove that `ax^(2) + bx + c` has at least one zero in (0, 1)
Let `ax(2) + bx + c ` is derivative of some function .
i.e., `f'(x) = ax^(2) + bx + c ` (1)
If we prove that `f(0) = f(1),` then from Rolle's theorem, there will lie at least one root of `f'(x) = 0 ` in (0, 1).
So , `f(x) = (ax^(3))/(3) + (bx^(2))/(2) + cx + d`
Now , `f(0) = d`
and `f(1) = (a)/(3) + (b)/(2) + c + d`
`(2a + 3b + 6c )/(6) + d`
= d (as 2a + 3b + 6c = 0)
Thus , f(0) = f(d).
Hence, from Rolle's theorem, there exitsts at least one root of
`f'(x) = 0 or ax^(2) + bx + c ` = 0 in ( 0, 1) .