Question

Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions:

`f(x)` = x² – x, x ∈ [0, 1]

Answer 1

f(0) = 0, f(1) = 0

⇒ f(0) = f(1) = 0

f(x) is continuous on [0, 1]

f(x) is differentiable on (0, 1)

Now, f'(x) = 2x – 1

Since, the tangent is parallel to x-axis then

f'(x) = 0

⇒ 2x – 1 = 0

x = `x/3` ∈ (0, 1)