state rolls theorem and verify Roll's theorem for the function f(x)=(x²-1)(x-2) on [-1,2]
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Answer:
The given function f(x)=x
2
(1−x
2
) on [0,1] is continuous and differentiable on (0,1), since it is a polynomial of degree 4.
Again f(0)=0=f(1).
So f(x) satisfies all the conditions of Rolle's theorem.
According to Rolle's theorem there exists c∈(0,1) such that f
′
(c)=0.
Now f
′
(c)=0 gives
2c−4c
3
=0
or, 2c(1−2c
2
)=0
or, c=
2
1
[ Since c∈(0,1)]
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