# What are the values and types of the critical points, if any, of #f(x)=cos^2 x - sin^2 x #?

We can rewrite as

Which has first derivative

The graph demonstrates:

graph{(cosx + sinx)(cosx - sinx) [-16.02, 16.02, -8.01, 8.01]}

Hopefully this helps!

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To find the critical points of ( f(x) = \cos^2 x - \sin^2 x ), first, we find its derivative.

( f'(x) = -2\sin x \cos x - (-2\sin x \cos x) = 0 )

The derivative equals zero when either ( \sin x = 0 ) or ( \cos x = 0 ). So, the critical points occur where ( \sin x = 0 ) or ( \cos x = 0 ).

When ( \sin x = 0 ), ( x = k\pi ), where ( k ) is an integer. When ( \cos x = 0 ), ( x = (2k+1)\frac{\pi}{2} ), where ( k ) is an integer.

So, the critical points are ( x = k\pi ) and ( x = (2k+1)\frac{\pi}{2} ), where ( k ) is an integer.

Now, to determine the type of each critical point, we can look at the behavior of the function around these critical points. We can do this by analyzing the signs of the derivative in intervals around each critical point.

At ( x = k\pi ), ( f'(x) = 0 ) and ( f''(x) = -2\sin^2 x - 2\cos^2 x = -2 < 0 ), so these are local maxima.

At ( x = (2k+1)\frac{\pi}{2} ), ( f'(x) = 0 ) and ( f''(x) = -2\sin^2 x - 2\cos^2 x = -2 < 0 ), so these are also local maxima.

Thus, the critical points of ( f(x) = \cos^2 x - \sin^2 x ) are all local maxima, and their values are ( f(k\pi) = 1 ) and ( f\left((2k+1)\frac{\pi}{2}\right) = -1 ), where ( k ) is an integer.

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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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