Is Rolle's theorem applicable to #f(x)=tanx#, when 0 < x < r??

Answer 1
No, since #f(0)!=f(r)# when #0 < r< pi/2#. But the Mean Value Theorem is applicable when #0 < r < pi/2#.
It says there is a number #c# (depending on #r#) in the interval #(0,r)# with the property that #sec^{2}(c)=\frac{tan(r)-tan(0)}{r-0}=tan(r)/r# (since #sec^{2}(x)=d/dx(tan(x))#), which is equivalent to #rsec^{2}(c)=tan(r)#.
One thing this implies is that, since #sec^{2}(c) = 1/(cos^{2}(c)) >1# when #0 < c < pi/2#, it follows that #tan(r)>r# when #0 < r < pi/2#. In other words, the graph of #y=tan(x)# is above the graph of #y=x# when #0 < x < pi/2#.
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Answer 2

Yes, Rolle's theorem is applicable to ( f(x) = \tan(x) ) on the interval ( 0 < x < r ), where ( r ) is any positive real number. Since ( \tan(x) ) is continuous and differentiable on the interval ( 0 < x < r ), and ( \tan(x) ) is continuous on the closed interval ( 0 \leq x \leq r ), and differentiable on the open interval ( 0 < x < r ), all the conditions required by Rolle's theorem are satisfied. Therefore, there exists at least one ( c ) in the open interval ( 0 < c < r ) such that ( f'(c) = 0 ).

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Answer 3

No, Rolle's theorem is not applicable to ( f(x) = \tan(x) ) on the interval ( 0 < x < r ), where ( r ) is any positive real number.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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