# How do you find all local maximum and minimum points using the second derivative test given #y=tan^2x#?

graph{(tanx)^2 [-7.09, 12.91, -2.12, 7.88]}

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To find all local maximum and minimum points using the second derivative test for (y = \tan^2(x)):

- Find the first derivative of (y) with respect to (x), (y').
- Find the critical points by solving (y' = 0) or (y') undefined.
- Compute the second derivative of (y) with respect to (x), (y'').
- Evaluate (y'') at each critical point.
- If (y'') is positive at a critical point, the point corresponds to a local minimum. If (y'') is negative at a critical point, the point corresponds to a local maximum. If (y'') equals zero, the test is inconclusive.

Given (y = \tan^2(x)):

- (y' = 2\tan(x)\sec^2(x))
- Critical points occur where (2\tan(x)\sec^2(x) = 0). These points are (x = k\pi) where (k) is an integer.
- (y'' = 2\sec^4(x) + 4\tan^2(x)\sec^2(x))
- Evaluate (y'') at each critical point.
- For (x = k\pi), (y'' = 2\sec^4(k\pi) + 4\tan^2(k\pi)\sec^2(k\pi)).

Since (\tan(k\pi) = 0) and (\sec(k\pi) = (-1)^k), we have:

- If (k) is even, then (\tan(k\pi) = 0) and (\sec(k\pi) = 1), so (y'' = 2). This implies a local minimum at (x = k\pi).
- If (k) is odd, then (\tan(k\pi) = 0) and (\sec(k\pi) = -1), so (y'' = -2). This implies a local maximum at (x = k\pi).

Therefore, all local maximum points occur at (x = (2n+1)\pi) for integer (n), and all local minimum points occur at (x = 2n\pi) for integer (n).

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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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