At what points on the following curves, is the tangent parallel to the

1.	At what points on the following curves, is the tangent parallel to the x–axis?y = x2 on [–2, 2]
Answer» 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 ⇒ \(\frac{d(x^2)}{dx}=0\) ⇒ 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).

At what points on the following curves, is the tangent parallel to the x–axis?y = x2 on [–2, 2]

Answer»

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

⇒ \(\frac{d(x^2)}{dx}=0\)

⇒ 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).