### Home > APCALC > Chapter 7 > Lesson 7.3.2 > Problem7-121

7-121.

Thoroughly investigate the graph of $y = \frac { 1 } { x }\cos(x)$ for $–4 ≤ x ≤ 4$. Identify all of the important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify point(s) of inflection and intercepts, and provide graphs of $f^\prime(x)$ and $f^{\prime\prime}(x)$. Be sure to justify all statements graphically and analytically.

To find INCREASING vs. DECREASING: Solve $f^\prime(x) > 0$ and $f ^\prime(x) < 0$.
To find CONCAVE UP vs. CONCAVE DOWN: Solve $f ^{\prime\prime}(x) > 0$ and $f^{\prime\prime}(x) < 0$.

LOCAL MAXIMA happen where increasing changes to decreasing.
LOCAL MINIMA happen where decreasing changes to increasing.
POINTS OF INFLECTION happen where concavity changes.

To find the GLOBAL MAXIMUM: compare $f\left(\text{endpoint}\right)$ with $f\left(\text{local maximum}\right)$. The highest value wins.
To find GLOBAL MINIMUM: compare $f\left(\text{endpoint}\right)$ with $f\left(\text{local minima}\right)$. The lowest value wins.