I'm asked to find max and min of: L(t)=12+2.8sin((2pi/365)(t-80).

I find the derivative as:
L'(t)=(5.6*pi/365)*cos[(2pi/365)(t-80)]

but I get lost afterwards.

You don't really need any derivatives for this. You know sin(z) has a max od 1 and a min of -1. So, when (2pi/365)(t-80) is an odd multiple of pi/2, sin is 1 or -1, so the min/max of L is 12±2.8

If you want to find the values of t where these extrema occur, then recall that max/min occur when L'=0. So, when does

cos[(2pi/365)(t-80)] = 0?

cos(z)=0 when z is an odd multiple of pi/2. So, for integer values of k, we need

(2pi/365)(t-80) = (2k+1)*pi/2
t-80 = (2k+1) pi/2 * 365/2pi
t-80 = 365(2k+1)/4
t = 80 + 365(2k+1)/4

t = 80±365/4, 80±1095/4, ...

To find the maximum and minimum values of a function, we need to find the critical points where the derivative is equal to zero or undefined.

In this case, the derivative of L(t) is given by:

L'(t) = (5.6*pi/365) * cos[(2pi/365)(t-80)]

To find the critical points, set the derivative equal to zero:

(5.6*pi/365) * cos[(2pi/365)(t-80)] = 0

Next, solve for t:

cos[(2pi/365)(t-80)] = 0

To find the values of t that satisfy this equation, we need to consider the values of cosine when it is equal to zero. The values of t that make cos[(2pi/365)(t-80)] equal to zero are the zeros of the cosine function, which occur at multiples of pi/2.

So, set the argument of the cosine function equal to pi/2:

(2pi/365)(t-80) = pi/2

Solve for t:

(t-80) = (365/(2pi))(pi/2)
t-80 = 365/4
t = 365/4 + 80
t = 272.5

Thus, the critical point is t = 272.5.

Now, let's analyze the behavior of the derivative around this critical point to determine whether it corresponds to a maximum or a minimum.

To do this, we can evaluate the sign of the derivative to the left and right of the critical point.

Evaluate L'(t) for t slightly less than 272.5:

L'(t) = (5.6*pi/365) * cos[(2pi/365)(t-80)]

L'(272) ≈ (5.6*pi/365) * cos[(2pi/365)(272-80)]

L'(272) ≈ (5.6*pi/365) * cos[(2pi/365)(192)]

Since the argument of the cosine function is positive, this means that cos[(2pi/365)(192)] is positive.

Therefore, L'(272) is positive.

Evaluate L'(t) for t slightly greater than 272.5:

L'(t) = (5.6*pi/365) * cos[(2pi/365)(t-80)]

L'(273) ≈ (5.6*pi/365) * cos[(2pi/365)(273-80)]

L'(273) ≈ (5.6*pi/365) * cos[(2pi/365)(193)]

Since the argument of the cosine function is negative, this means that cos[(2pi/365)(193)] is negative.

Therefore, L'(273) is negative.

Now, let's analyze the behavior of the function L(t) around the critical point to determine whether it corresponds to a maximum or a minimum.

Evaluate L(t) for t slightly less than 272.5:

L(272) = 12+2.8*sin[(2pi/365)(272-80)]

Evaluate L(t) for t slightly greater than 272.5:

L(273) = 12+2.8*sin[(2pi/365)(273-80)]

By comparing the values of L(272) and L(273), we can see that L(273) is smaller than L(272).

Therefore, the critical point t = 272.5 corresponds to a maximum value of the function L(t).

To summarize:
- The maximum value of the function L(t) is obtained at t = 272.5.
- The minimum value of the function L(t) is not specified in the given information.

To find the maximum and minimum of a given function, you need to analyze the behavior of its derivative. In this case, you have correctly calculated the derivative of L(t) as L'(t) = (5.6π/365) * cos[(2π/365)(t-80)].

To find the maximum and minimum points, you need to first set the derivative equal to zero to find the critical points. In other words, find the values of t for which L'(t) = 0.

Setting L'(t) = 0, we have:
(5.6π/365) * cos[(2π/365)(t-80)] = 0

To solve this equation, we need to find the values of t for which cos[(2π/365)(t-80)] = 0.

The cosine function equals zero at angles of (2n+1)π/2, where n is an integer. Therefore, we have:

(2π/365)(t-80) = (2n+1)π/2

Simplifying the equation, we get:
(t-80) = [(2n+1)/2] * (365/π)

Now, solve for t:
t = 80 + [(2n+1)/2] * (365/π)

These values of t represent the critical points where the derivative is zero. To determine whether they correspond to maximum or minimum points, you need to examine the behavior of the derivative on either side of these critical points.

Calculate the second derivative of L(t) to determine the nature of these critical points. The second derivative is obtained by differentiating the derivative L'(t):

L''(t) = -(2π/365)*(5.6π/365)*sin[(2π/365)(t-80)]

Now, evaluate L''(t) at each critical point (t-values obtained earlier). If L''(t) > 0, then the t-value corresponds to a minimum point. If L''(t) < 0, then the t-value corresponds to a maximum point. If L''(t) = 0, then the test is inconclusive.

When analyzing the behavior of L''(t), consider the range of t-values that are of interest to you (for example, if you're interested in the behavior of L(t) over a specific time period).

By applying this method, you can determine the maximum and minimum points for the given function L(t).