differentiation

(a) First , it pass through (0,1) ,
so , c = 1 .

Second , it touches the x-axis at x=1
means that it pass through (1,0) and
dy/dx =0 when x=1.

Then we get , 1+a+b+c = 0 => a + b + 2 =0 ..(1)

also dy/dx = 3x^2 + 2ax + b = 0 when x=1
Then we get 2a + b + 3 = 0 ...(2)

On solving , a = -1 and b = -1
Therefore , a = b = -1 and c = 1.

(b) dy/dx = 3x^2 + 2ax + b = 3x^2 - 2x - 1
Setting dy/dx = 0 ,
3x^2 - 2x -1 =0
(x-1)(3x+1) =0
x = 1 or -1/3
for x = -1/3 , y = -1/27 - 1/9 + 1/3 + 1 = 32/27
So turning pts are ( -1/3 , 32/27) and ( 1,0)