Please help me!
Find the formula for the function of the form w(x)= Asin(Bx)+C with an (i) a maximum (3,6),(i) a minimum at (-9,6), and (iii) no critical points between these two points
how can it have a max and a min at y=6, and no critical points in between?
By Rolle's Theorem, there must be a point in the interval where w'(x) = 0
But that is another critical point.
And anyway, since the amplitude A is 1/2 the difference between the max and the min, A=0, right?
But the proposed solution is bogus, since
If w(x) = 6 sin(pi/6 x) + 0 then
w(3) = 6 * 1/2 + 0 = 3
so let's say that there is a maximum at (3,6) and a minimum at (-9,-6)
The A = (6+6)/2 = 6 and C = (6-6)/2 = 0
But now you have to have a phase shift, so that means that since the period is 2(3+9) = 24
w(x) = 6cos(pi/12 (x-3)) = 6sin(pi/12 (x+3))
6 sin(pi/6 x) +0
To find the formula for the function w(x) = Asin(Bx) + C given the conditions, we can use the following steps:
Step 1: Maximum point at (3, 6)
When the function has a maximum point at (3, 6), we can determine the value of B using the relationship between the period of the sine function and B. The period, P, is related to B as follows: P = 2π/B.
In this case, the maximum point occurs at x = 3. Since the maximum point corresponds to half a period, we can set up an equation as follows:
3 = P/2 = 2π/B
Solving for B:
2π/B = 3
B = 2π/3
Step 2: Minimum point at (-9, 6)
When the function has a minimum point at (-9, 6), we can determine the value of B using the same approach. Since the minimum point corresponds to half a period, we set up the equation as:
-9 = P/2
2π/B = -9
Solving for B:
2π/B = -9
B = -2π/9
Step 3: No critical points between (-9, 6) and (3, 6)
To ensure there are no critical points between these two points, we need to make sure the period of the function is less than or equal to the distance between them. The distance between these points is 3 - (-9) = 12.
The period, P, is related to B as P = 2π/B. So, we need P ≤ 12:
2π/B ≤ 12
Substituting the values of B from steps 1 and 2:
2π/(2π/3) ≤ 12
3 ≤ 12
Since 3 is indeed less than or equal to 12, the condition of no critical points between (-9, 6) and (3, 6) is satisfied.
Therefore, the formula for the function w(x) is:
w(x) = Asin(Bx) + C
where A is the amplitude, B = 2π/3, and C is the vertical shift (3, 6) being the maximum point.