Task B-3.5. The function $f(x)=2 \cos 2x + 4 \cos x + 3$ is given. Determine the set of all values that the function $f$ can take.
## Solution.

We are looking for the minimum and maximum value of the function $f$.

Let's write the function $f$ in a more convenient form

$$
f(x)=2 \cos 2 x+4 \cos x+3=2\left(2 \cos ^{2} x-1\right)+4 \cos x+3=(2 \cos x+1)^{2}
$$

The function $f$ will have its maximum value when the cosine has its maximum value, i.e., when $\cos x=1$. Therefore, the maximum is $(2+1)^{2}=9$.

(1 point)

The minimum is obviously 0 because $f(x) \geq 0$ for all $x \in \mathbb{R}$, and $f(x)=0$ when $\cos x=-0.5$.

(Or we can look at it as the x-coordinate of the vertex of the graph of the quadratic function.)

Conclusion $f(x) \in[0,9]$.