The slope of the tangent to a curve f(x) at any point x = c is the value of f'(c)

We have to determine a and b for which the parabola y = x^2 + ax + b and the curve y = x^3 have the same tangent at the point (1,1).

The slope of the tangent y = x^3 at (1,1) is y' = 3x^2 at x = 1. The slope is 3.

The value of the derivative y' = 2x + a. At x = 1 y' = 2 + a.

If 2 + a = 3

=> a = 1

At (1,1)

1 = 1 + a + b

=> 1 = 1 + 1 + b

=> b = -1

The graphs showing the curves y = x^3 and y = x^2 + x - 1 clearly shows that the two curves have the same tangent at (1,1)

The required value of a = 1 and b = -1