I'll help you begin. In order for the headligts to shine on the sign the slope of the road dy/dx must equal the slope of a line conecting the car to the sign. So given that the sign is at coordinates (A,B), the car is at (x,y), and y = x^2, you have:
Now you can solve for x, then y.
As for part (b) the car is at its closest approach when the slope of the line connecting the car to the sign is the negative inverse of the slope of the road:
For (c) consider the chain rule.