When does this trig function have its maximum value?

When does this trig function have its maximum value?

Question

The greatest value which the function $f(x)=(3\sin^2(10x+11)-7)^2$ takes, as $x$ varies over all real values, equals

1. $-9$,
2. $16$,
3. $49$,
4. $100$.

Solution

We know that $\sin(10x+11)$ takes values between $-1$ and $1$.

So $\sin^2(10x+11)$ takes values between $0$ and $1$

So $3\sin^2(10x+11)$ takes values between $0$ and $3$

So $3\sin^2(10x+11)-7$ takes values between $-7$ and $-4$

So $f(x)$ takes values between $49$ and $16$, and the greatest value which $f(x)$ takes is $49$.

The answer is c.