First, let us write the conditions for the applicability of Rolle’s theorem:

For a Real valued function ‘f’:

a) The function ‘f’ needs to be continuous in the closed interval [a,b].

b) The function ‘f’ needs differentiable on the open interval (a,b).

c) f(a) = f(b)

Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.

Given function is:

⇒ y = x² on [ – 2,2]

We know that polynomials are continuous and differentiable over R.

Let’s check the values of y at the extremums

⇒ y( – 2) = ( – 2)²

⇒ y( – 2) = 4

⇒ y(2) = (2)²
⇒ y(2) = 4

We got y( – 2) = y(2). So, there exists a c such that f’(c) = 0.

For a curve g to have a tangent parallel to x – axis at point r, the criteria to be satisfied is g’(r) = 0.

⇒ y’(x) = 0

⇒

⇒ 2x = 0

⇒ x = 0

The value of y is

⇒ y = (0)²

⇒ y = 0

The point at which the curve has tangent parallel to x – axis is (0,0).