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) = sqrt(x) - x/3, x ∈ [0, 9]`

Answer 1

f(x) = `sqrt(x) - x/3, x ∈ [0, 9]`

f(0) = 0, f(9) = `sqrt(9) - 9/3` = 3 – 3 = 0

⇒ f(0) = 0 = f(9)

f(x) is continuous on [0, 9]

f(x) is differentiable on (0, 9)

Now f'(x) = `1/(2sqrt(x)) - 1/3`

Since, the tangent is parallel to x-axis.

f'(x) = 0

`1/(2sqrt(x)) - 1/3` = 0

⇒ `1/(2sqrt(x)) = 1/3`

`sqrt(x) = 3/2`

x = `9/4`

x = `9/4 ∈ (0, 9)`