## Task B-3.4.

Happy Stef, returning from a friend's birthday party, was walking on a path that can be described by the graph of the function $f(x)=2 \sin ^{2} x+2 \sin x+1$. How wide is the narrowest bridge without a railing that Stef successfully crossed in its entirety and reached the other side? (The $x$-axis follows the direction of the bridge from one bank to the other, and the $y$-axis follows the direction of the river flow.)
## Solution.

The width of the bridge is equal to the difference between the minimum and maximum values of the function $f$.

The function $f$ attains its maximum value when $\sin x=1$, and for these values, it is 5.

On the other hand,

$$
f(x)=2 \sin ^{2} x+2 \sin x+1=2\left[\left(\sin x+\frac{1}{2}\right)^{2}-\frac{1}{4}\right]+1=2\left(\sin x+\frac{1}{2}\right)^{2}+\frac{1}{2} . \quad 1 \text { point }
$$

For $\sin x=-\frac{1}{2}$, the function $f$ attains its minimum value, which is $\frac{1}{2}$.

1 point

The bridge is at least $5-\frac{1}{2}=4.5$ meters wide.

Note: The minimum could also be determined by the substitution $t=\sin x$ and finding the minimum of the quadratic function, with the comment that the $t$ for which the function achieves its minimum is indeed among the values $[-1,1]$ that $\sin x$ attains.