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
that's incorrect
To find the formula for the function of the form w(x) = Asin(Bx) + C with the given conditions, we need to determine the values of A, B, and C.
(i) Maximum at (3,6):
When x = 3, w(x) = 6 (maximum value)
So, we have the equation 6 = Asin(B(3)) + C.
(ii) Minimum at (-9,6):
When x = -9, w(x) = 6 (minimum value)
So, we have the equation 6 = Asin(B(-9)) + C.
(iii) No critical points between these two points:
To have no critical points between these two points, we need to ensure that the derivative of w(x) does not equal zero between x = -9 and x = 3.
The derivative of w(x) is given by w'(x) = ABcos(Bx).
To find the values of A, B, and C, we'll solve the equations:
Equation 1: 6 = Asin(3B) + C
Equation 2: 6 = Asin(-9B) + C
Equation 3: w'(x) ≠ 0, for -9 ≤ x ≤ 3
Solving Equation 1 and Equation 2 for A and C:
Equation 1 - Equation 2: Asin(3B) - Asin(-9B) = C - C
2Asin(3B) = 0
sin(3B) = 0
B = (2nπ)/3 (where n is an integer)
Since B cannot be zero, we have three possible values for B: B = (2π)/3, B = (4π)/3, and B = (6π)/3.
Now, substituting each value of B into Equation 1 and Equation 2, we can solve for A and C.
For B = (2π)/3:
6 = Asin((2π)/3 * 3) + C
6 = Asin(2π) + C
6 = Asin(0) + C
6 = 0 + C
C = 6
For B = (4π)/3:
6 = Asin((4π)/3 * 3) + C
6 = Asin(4π) + C
6 = Asin(0) + C
6 = 0 + C
C = 6
For B = (6π)/3:
6 = Asin((6π)/3 * 3) + C
6 = Asin(6π) + C
6 = Asin(0) + C
6 = 0 + C
C = 6
Hence, we can conclude that C = 6 for all values of B.
Finally, the formula for the given conditions is:
w(x) = A*sin(Bx) + 6,
where A can be any real number, and B can take the values (2π)/3, (4π)/3, or (6π)/3.
w(x) = 6 by Rolle's Theorem.
I suspect a typo.