Question

using rolle's theorem find the point on y = x(x-4) , x belongs to [0,4] where the tangent is parallel to x-axis

Answer 1

Clearly

f(x)=x(x-4)
=x²-4x

The function is continuous on [0,4] and differentiable on (0,4).

f(0)=0
f(4)=0

All the conditione for Rolles theorem are satisfied.
Consequently, there exist atleast one point which belongs to (0,4) for which f¹(c) =0.

f¹(x) =2x-4 =0
f¹(c) =2c-4 =0
c=2.

f(c)=-4.


By geometrical representation of Rolles theorem, (-4,2) is the point on y=x(x-4),where tangent is parallel to the x axis.