The quantity demanded each month of the Sicard wristwatch is related to the unit price given by the equation below, where p is measured in dollars and x is measured in units of a thousand. To yield a maximum revenue, how many watches must be sold? (Round your answer to the nearest whole number.)

P=41/0.01X^2+1 (0(less than or equal to)x(less than or equal to)20)

? watches

since P(x) is a decreasing function, maximum value is attained at the minimum x in the domain: 0.

To determine the number of watches that must be sold to yield maximum revenue, we need to find the value of x that maximizes the equation P = (41/0.01) x^2 + 1.

To find the maximum value, we can take the derivative of the equation and set it equal to zero. Let's do that step by step:

1. Start with the equation: P = (41/0.01) x^2 + 1
2. Take the derivative with respect to x: dP/dx = 2 (41/0.01) x
3. Set the derivative equal to zero and solve for x: 2 (41/0.01) x = 0
(41/0.01) x = 0
x = 0

Now we have found that x = 0 is a critical point, but we need to check if it is a maximum or minimum. To do this, we can use the second derivative test.

4. Take the second derivative with respect to x: d²P/dx² = 2 (41/0.01)
d²P/dx² = 2 (4100)
d²P/dx² = 8200

Since the second derivative is positive, we can conclude that x = 0 is a local minimum, and since the given interval is [0, 20], it is also the absolute minimum. Therefore, we need to find the maximum value on this interval.

Next, we evaluate the equation for the endpoints of the interval, x = 0 and x = 20.

5. P(0) = (41/0.01) (0)^2 + 1 = 1
6. P(20) = (41/0.01) (20)^2 + 1 = 16501

Therefore, on the interval [0, 20], the maximum value of P is 16501 which occurs when x = 20.

To find the number of watches that must be sold to yield maximum revenue (P), we need to convert x from units of a thousand to units of 1. Therefore, we multiply x by 1000.

So, the number of watches that must be sold to yield maximum revenue is 20,000 (rounded to the nearest whole number).