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

Thoroughly investigate the graph of *y* = *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* ′(*x*) and *f* ″(*x*). Be sure to justify all statements *graphically *and *analytically. *Homework Help ✎

To find INCREASING vs. DECREASING: Solve *f* '(*x*) > 0 and *f* '(*x*) < 0.

To find CONCAVE UP vs. CONCAVE DOWN: Solve *f* ''(*x*) > 0 and *f* ''(*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.