Is #f(x)=cos^2x+sin2x# increasing or decreasing at #x=pi/6#?
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To determine if ( f(x) = \cos^2(x) + \sin(2x) ) is increasing or decreasing at ( x = \frac{\pi}{6} ), we need to evaluate the derivative of ( f(x) ) at that point. The derivative of ( f(x) ) is ( f'(x) = 2\sin(x)\cos(x) + 2\cos(2x) ). Evaluating this derivative at ( x = \frac{\pi}{6} ), we get ( f'(\frac{\pi}{6}) = 2\sin(\frac{\pi}{6})\cos(\frac{\pi}{6}) + 2\cos(\frac{\pi}{3}) ). Simplifying this expression, we find ( f'(\frac{\pi}{6}) = \sqrt{3} + 1 ). Since ( f'(\frac{\pi}{6}) ) is positive, ( f(x) ) is increasing at ( x = \frac{\pi}{6} ).
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To determine if the function ( f(x) = \cos^2(x) + \sin(2x) ) is increasing or decreasing at ( x = \frac{\pi}{6} ), we need to examine the derivative of the function at that point.
 Find the derivative of the function ( f(x) ).
 Evaluate the derivative at ( x = \frac{\pi}{6} ).
 Determine the sign of the derivative to ascertain if the function is increasing or decreasing at ( x = \frac{\pi}{6} ).
Let's calculate:

Derivative of ( f(x) ): [ f'(x) = 2\cos(x)\sin(x) + 2\cos(2x) ]

Evaluate ( f'(x) ) at ( x = \frac{\pi}{6} ): [ f'\left(\frac{\pi}{6}\right) = 2\cos\left(\frac{\pi}{6}\right)\sin\left(\frac{\pi}{6}\right) + 2\cos\left(\frac{\pi}{3}\right) ] [ = 2\left(\frac{\sqrt{3}}{2}\right)\left(\frac{1}{2}\right) + 2\left(\frac{1}{2}\right) ] [ = \sqrt{3} + 1 ]

Since ( f'\left(\frac{\pi}{6}\right) = \sqrt{3} + 1 ) is positive, the function ( f(x) ) is increasing at ( x = \frac{\pi}{6} ).
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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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