How do you verify whether rolle's theorem can be applied to the function #f(x)=absx# in [1,1]?
According to Rolle's Theorem, there are three criteria for us to able to apply the theorem:
1.)
2.)
3.)
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To verify whether Rolle's Theorem can be applied to the function ( f(x) = x ) in the interval ([1, 1]), we need to check two conditions:

Continuity: Verify if ( f(x) = x ) is continuous on the closed interval ([1, 1]). Since ( x ) is continuous everywhere, including at ( x = 1 ) and ( x = 1 ), it is continuous on the interval ([1, 1]).

Differentiability: Check if ( f(x) = x ) is differentiable on the open interval ((1, 1)). At ( x = 0 ), the derivative of ( f(x) = x ) does not exist since the function has a sharp corner at ( x = 0 ). However, for ( x \neq 0 ), the function is differentiable as it behaves like the function ( y = x ) or ( y = x ). Thus, the function is differentiable on the open interval ((1, 1) \setminus {0}).
Since both conditions are met (continuity on ([1, 1]) and differentiability on ((1, 1) \setminus {0})), Rolle's Theorem can be applied to the function ( f(x) = x ) in the interval ([1, 1]).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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