Q.9. This question is about pairs of functions $f$ and $g$ that satisfy

$$f(x) - g(x) = 2 \sin x$$

$$f(x) g(x) = \cos^2 x$$

for all real numbers $x$.

Across all solutions for $f(x)$, what is the minimum value that $f(x)$ attains for any $x$?

Answer Choices

• $1 - \sqrt{2}$
• $-1 - \sqrt{2}$
• $0$
• $-1$
• $-2$
• $-3$
• $-4$

Solution

We substitute $f(x) = g(x) + 2 \sin x$ into the second equation and solve the resulting quadratic equation for $g(x)$. This gives two possible expressions for $g(x)$, which lead to two possible expressions for $f(x)$: $f(x) = \sin x + 1$ and $f(x) = \sin x - 1$.

The minimum value of $f(x)$ is $-2$, which occurs when $f(x) = \sin x - 1$ and $\sin x = -1$.

The correct answer is: $-2$.