Verify Rolle’s theorem for each of the following functions:
Condition (1):
Since, f(x)=x(x-4)2 is a polynomial and we know every polynomial function is continuous for all xϵR.
⇒ f(x)= x(x-4)2 is continuous on [0,4].
Condition (2):
Here, f’(x)= (x-4)2+2x(x-4) which exist in [0,4].
So, f(x)= x(x-4)2 is differentiable on (0,4).
Condition (3):
Here, f(0)=0(0-4)2=0
And f(4)= 4(4-4)2=0
i.e. f(0)=f(4)
Conditions of Rolle’s theorem are satisfied.
Hence, there exist at least one cϵ(0,4) such that f’(c)=0
i.e. (c-4)2+2c(c-4)=0
i.e. (c-4)(3c-4)=0
i.e. c=4 or c=3÷4
Value of
Thus, Rolle’s theorem is satisfied.