How do you find all local maximum and minimum points using the second derivative test given #y=cos^2xsin^2x#?
As:
However we can check:
so the critical points occur for:
and in fact:
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To find the local maximum and minimum points using the second derivative test for ( y = \cos^2(x)  \sin^2(x) ):

Find the first derivative, ( y' ). [ y' = \frac{d}{dx}(\cos^2(x)  \sin^2(x)) = 2\cos(x)\sin(x) ]

Find the critical points by setting the first derivative equal to zero and solving for ( x ). [ 2\cos(x)\sin(x) = 0 ] [ \Rightarrow \cos(x) = 0 \text{ or } \sin(x) = 0 ]

Solve for ( x ) for each condition: [ \cos(x) = 0 \Rightarrow x = \frac{\pi}{2} + n\pi, \text{ where } n \in \mathbb{Z} ] [ \sin(x) = 0 \Rightarrow x = n\pi, \text{ where } n \in \mathbb{Z} ]

Calculate the second derivative, ( y'' ). [ y'' = \frac{d}{dx}(2\cos(x)\sin(x)) = 2(\sin^2(x)  \cos^2(x)) ]

Evaluate ( y'' ) at each critical point: [ \text{For } x = \frac{\pi}{2} + n\pi, \text{ where } n \in \mathbb{Z}: ] [ y'' = 2(\sin^2(\frac{\pi}{2} + n\pi)  \cos^2(\frac{\pi}{2} + n\pi)) ] [ \text{For } x = n\pi, \text{ where } n \in \mathbb{Z}: ] [ y'' = 2(\sin^2(n\pi)  \cos^2(n\pi)) ]

Determine the nature of the critical points using the second derivative test:
 If ( y'' > 0 ), the critical point is a local minimum.
 If ( y'' < 0 ), the critical point is a local maximum.
 If ( y'' = 0 ), the test is inconclusive.
 Plug in the critical points into the second derivative and determine the nature of each critical point.
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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.
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