Question

Using Rolle’s theorem, find the point on the curve y = x(x - 4), x ∈ [0, 4], where the tangent is parallel to the x -axis.

Answer 1

Condition (1):

Since, y = x(x - 4) is a polynomial and we know every polynomial function is continuous for all x ϵ R.

⇒ y = x(x - 4) is continuous on [0,4].

Condition (2):

Here, y’= (x - 4)+x which exist in [0,4].

So, y = x(x - 4) is differentiable on (0,4).

Condition (3):

Here, y(0) = 0(0 - 4) = 0

And y(4) = 4(4 - 4) = 0

i.e. y(0) = y(4)

Conditions of Rolle’s theorem are satisfied.

Hence, there exist at least one c ϵ (0,4) such that y’(c) = 0

i.e. (c - 4)+c = 0

i.e. 2c - 4 = 0

i.e. c = 2

Value of c = 2 ϵ (0,4)

So, y(c) = y(2) = 2(2 - 4) = - 4

By geometric interpretation, (2,- 4) is a point on a curve y = x(x - 4),where tangent is parallel to x -axis.