
Verify Rolle’s Theorem for the function where
Answer
529.2k+ views
Hint: Rolle’s Theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a)=f(b), then for some x, .
Given function is and x belongs to [-2, 2]
Here y is a function of x
The given function f(x) is continuous on a closed interval [-2, 2] and differentiable on an open interval (-2, 2).
We have f (2) = f (-2) = 6
According to Rolle’s Theorem, if f (-2) = f (2) then there exists at least one point c in (-2, 2) such that
Now, to check whether such c exists or not
We have
for x = 0, and -2 < 0 < 2
Hence, there exist such that
Therefore, Rolle’s Theorem is verified.
Note:While verifying Rolle’s Theorem for a function, we need to make sure that it is satisfying all the three rules of Rolle’s Theorem. If any of those conditions failed then, we can say that Rolle’s Theorem is not applicable for that function.
Function is continuous means its graph is unbroken without any holes (discontinuity). Function must be defined at every point of the given interval.
Given function is
Here y is a function of x
The given function f(x) is continuous on a closed interval [-2, 2] and differentiable on an open interval (-2, 2).
We have f (2) = f (-2) = 6
According to Rolle’s Theorem, if f (-2) = f (2) then there exists at least one point c in (-2, 2) such that
Now, to check whether such c exists or not
We have
Hence, there exist
Therefore, Rolle’s Theorem is verified.
Note:While verifying Rolle’s Theorem for a function, we need to make sure that it is satisfying all the three rules of Rolle’s Theorem. If any of those conditions failed then, we can say that Rolle’s Theorem is not applicable for that function.
Function is continuous means its graph is unbroken without any holes (discontinuity). Function must be defined at every point of the given interval.
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