The maximum value of 12 sin(theta) - 9 sin ^2 (theta) is
a) 3 b) 4
c) 5 d) none of these
Dear Student,
Let z = sin x
Then
y = 12 sin (x)- 9 sin^2 (x)
becomes
y = 12z - 9z^2 = 3z(4 - 3z)
which has roots at 0 and 4/3
The vertex is half-way between 0 and 4/3:
z = (0 + 4/3)/2 = 2/3
This is good because sin x = 2/3 has a
solution. We don't need to know what
it is, but we need to know that it exists.
The maximum value occurs at the vertex:
y_max = 3(2/3)(4 - 3(2/3)) = 2(4 - 2) = 4
Regards
Let z = sin x
Then
y = 12 sin (x)- 9 sin^2 (x)
becomes
y = 12z - 9z^2 = 3z(4 - 3z)
which has roots at 0 and 4/3
The vertex is half-way between 0 and 4/3:
z = (0 + 4/3)/2 = 2/3
This is good because sin x = 2/3 has a
solution. We don't need to know what
it is, but we need to know that it exists.
The maximum value occurs at the vertex:
y_max = 3(2/3)(4 - 3(2/3)) = 2(4 - 2) = 4
Regards