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

- \(-9\),
- \(16\),
- \(49\),
- \(100\).

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.