Q. 8
Verify Rolle’s theorem for each of the following functions:

Answer :
Condition (1):
Since, f(x)= x3+3x2-24x-80 is a polynomial and we know every polynomial function is continuous for all xϵR.
⇒ f(x)= x3+3x2-24x-80 is continuous on [-4,5].
Condition (2):
Here, f’(x)= 3x2+6x-24 which exist in [-4,5].
So, f(x)= x3+3x2-24x-80 is differentiable on (-4,5).
Condition (3):
Here, f(-4)= (-4)3+3(-4)2-24(-4)-80=0
And f(5)= (5)3+3(5)2-24(5)-80=0
i.e. f(-4)=f(5)
Conditions of Rolle’s theorem are satisfied.
Hence, there exist at least one cϵ(-4,5) such that f’(c)=0
i.e. 3c2+6c-24=0
i.e. c=-4 or c=2
Value of c=2 ϵ(-4,5)
Thus, Rolle’s theorem is satisfied.
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