CPM Homework Help : CALC Problem 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.

Step 1:

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 $f^{\prime\prime}(x) < 0$ .

Step 2:

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

Step 3:

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.