## SUBJECT IV

Consider the function $f:\left[\frac{\pi}{3}, \frac{4 \pi}{3}\right] \rightarrow[-2,2]$ given by the formula $f(x)=\cos x+\sqrt{3} \sin x$.

a) Determine the constants $a$ and $b$ such that $f(x)=a \cos (x+b)$.

b) Represent the function $f$ graphically.

c) Show that $f$ is invertible.

## Note:

All subjects are mandatory.

## Working time 3 hours.

Each subject is graded from 0 to 7.

## MATHEMATICS OLYMPIAD  LOCAL STAGE - FEBRUARY 16, 2013  10th Grade  GRADING AND SCORING GUIDE

- For any correct solution, even if it is different from the one in the guide, the corresponding score is given.
- No fractional points are awarded, but intermediate scores can be given for partial solutions, within the limits of the score indicated in the guide.
SUBIECTUL IV

| a) | $f(x)=2 \cos \left(x-\frac{\pi}{3}\right)$, so $a=2$ and $b=-\frac{\pi}{3}$ | $\mathbf{2 p}$ |
| :--- | :--- | :--- |
| b) | $A\left(\frac{\pi}{3}, 2\right), B\left(\frac{5 \pi}{6}, 0\right), C\left(\frac{4 \pi}{3},-2\right)$ are the three notable points | $\mathbf{2 p}$ |
| c) | Proof of injectivity  Proof of surjectivity  Conclusion | $\mathbf{1 p}$ |