<h2>Answer with explanation:</h2>
Rolle's Theorem states that:
If f is a continuous function in [a,b] and is differentiable in (a,b)
such that f(a)=f(b)
Then there exist a constant c in between a and b i.e. c∈[a,b]
such that: f'(c)=0
Here we have the function f(x) as:
where x∈[-1,3]
- Since the function f(x) is a polynomial function hence it is continuous as well as differentiable over the interval [-1,3].
Also,
f(-1)=15
(Since,
)
and f(3)=15
( Since,
)
- i.e. f(-1)=f(3)
Hence, there will exist a c∈[-1,3] such that f'(c)=0
Hence, the c that satisfy the conclusion is: c=1