2. (16 points) Find the minimum value of the function

$$
f(x)=7 \sin ^{2} x+5 \cos ^{2} x+2 \sin x
$$
Answer: 4.5.

Solution. Using the basic trigonometric identity, we get $f(x)=$ $2 \sin ^{2} x+2 \cos x+5$. Let's make the substitution $\sin x=t, t \in[-1 ; 1]$. After this, the problem reduces to finding the minimum value of the quadratic function $g(x)=2 t^{2}+$ $2 t+5$ on the interval $[-1 ; 1]$. The quadratic function defines a parabola on the plane (with branches directed upwards), the minimum value is achieved at the vertex of the parabola, provided the abscissa of the vertex belongs to the interval $[-1 ; 1]$. The abscissa of the vertex of the parabola $t_{0}=-\frac{1}{2} \in[-1 ; 1]$, therefore, $g_{\text {min }}=g\left(-\frac{1}{2}\right)=\frac{1}{2}-1+5=4.5$.