Find the Maximum Value of 12 sin x − 9 sin²x
Question:
\[
12\sin x-9\sin^2x
\]
Write its maximum value.
Solution
Let \[ \sin x=t \] where \[ -1\le t\le1 \]
Then, \[ 12\sin x-9\sin^2x = 12t-9t^2 \]
\[ = -9\left(t^2-\frac43 t\right) \]
\[ = -9\left[\left(t-\frac23\right)^2-\frac49\right] \]
\[ = -9\left(t-\frac23\right)^2+4 \]
Since \[ \left(t-\frac23\right)^2\ge0 \] maximum value occurs when \[ \left(t-\frac23\right)^2=0 \]
\[ \boxed{4} \]