E. MAXIMUM, MINIMUM, AND INFLECTION POINTS:
CURVE SKETCHING
Case I. Functions That Are Everywhere Differentiable
The following procedure is suggested to determine any maximum, minimum, or
inflection point of a curve and to sketch the curve.
(1) Find y′ and y′′.
(2) Find all critical points of y, that is, all x for which y′ = 0. At each of these x’s
the tangent to the curve is horizontal.
(3) Let c be a number for which y′ is 0; investigate the sign of y′′ at c. If y′′(c) >
0, then c yields a local minimum; if y′′(c) < 0, then c yields a local
maximum. This procedure is known as the Second Derivative Test (for
extrema). See Figure N4–2. If y′′(c) = 0, the Second Derivative Test fails
and we must use the test in step (4) below.