How do you verify whether rolle's theorem can be applied to the function #f(x)=1/x^2# in [1,1]?
Rolle's Theorem states that if a function,
So what Rolle's Theorem is stating should be obvious as if the function is differentiable then it must be continuous (as differentiability
With
Differentiating wrt
To find a turning point we require;
# f'(x)=0 => (2)/x^3 = 0#
Which has no finite solution. We can conclude that
We can see that this is the case graphically, as f(x) has a discontinuity when
graph{1/x^2 [10, 10, 2, 10]}
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To verify whether Rolle's Theorem can be applied to the function (f(x) = \frac{1}{x^2}) in the interval ([1, 1]), follow these steps:

Check if the function is continuous on the closed interval ([1, 1]). Since (f(x) = \frac{1}{x^2}) is continuous everywhere on its domain except at (x = 0), and ([1, 1]) does not include (x = 0), the function is continuous on ([1, 1]).

Check if the function is differentiable on the open interval ((1, 1)). Since (f(x) = \frac{1}{x^2}) is differentiable everywhere on its domain except at (x = 0), and ([1, 1]) does not include (x = 0), the function is differentiable on ((1, 1)).

Verify if (f(1) = f(1)). Evaluate (f(1)) and (f(1)). If (f(1) = f(1)), then Rolle's Theorem can be applied.
Let's check: [f(1) = \frac{1}{(1)^2} = 1] [f(1) = \frac{1}{(1)^2} = 1]
Since (f(1) = f(1)), Rolle's Theorem can be applied to the function (f(x) = \frac{1}{x^2}) 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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