Using Rolle’s theorem, find the point on the curve , where the tangent is parallel to the x-axis.
Condition (1):
Since, y=x(x-4) is a polynomial and we know every polynomial function is continuous for all xϵR.
⇒ 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 cϵ(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=2ϵ(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.