# How do you determine if rolles theorem can be applied to # f(x) = sin 2x# on the interval [0, (pi/2)] and if so how do you find all the values of c in the interval for which f'(c)=0?

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To determine if Rolle's Theorem can be applied to ( f(x) = \sin(2x) ) on the interval ( [0, \frac{\pi}{2}] ), check if the following conditions are met:

- The function ( f(x) ) is continuous on the closed interval ( [0, \frac{\pi}{2}] ).
- The function ( f(x) ) is differentiable on the open interval ( (0, \frac{\pi}{2}) ).
- ( f(0) = f\left(\frac{\pi}{2}\right) ).

If all these conditions are satisfied, then Rolle's Theorem can be applied. To find all the values of ( c ) in the interval ( [0, \frac{\pi}{2}] ) for which ( f'(c) = 0 ), differentiate ( f(x) ) with respect to ( x ), then set the derivative equal to zero and solve for ( c ).

For ( f(x) = \sin(2x) ): [ f'(x) = 2\cos(2x) ]

Setting ( f'(x) = 0 ) and solving for ( x ), we get: [ 2\cos(2c) = 0 ] [ \cos(2c) = 0 ]

This equation has solutions in the interval ( [0, \frac{\pi}{2}] ) when ( 2c = \frac{\pi}{2} ) or ( 2c = \frac{3\pi}{2} ).

So, the values of ( c ) where ( f'(c) = 0 ) in the interval ( [0, \frac{\pi}{2}] ) are: [ c_1 = \frac{\pi}{4} ] [ c_2 = \frac{3\pi}{4} ]

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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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