  ### Home > CALC > Chapter 7 > Lesson 7.4.2 > Problem7-186

7-186.

Thoroughly investigate the graph of $y =\frac { 1 } { x }\operatorname{cos} x$ for $−4 ≤ x ≤ 4.$ Identify all 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 $f$(endpoint) with $f$(local maximum). The highest value wins.
To find GLOBAL MINIMUM: compare $f$(endpoint) with $f$(local minima). The lowest value wins.