proper concavity.
Example 12 __
Find any maximum, minimum, or inflection points on the graph of f (x) = x^3 −
5 x^2 + 3 x + 6, and sketch the curve.
SOLUTION: For the steps listed above:
(1) Here f ′(x) = 3x^2 − 10x + 3 and f ′′(x) = 6x − 10.
(2) f ′(x) = (3x − 1)(x − 3), which is zero when x = or 3.
(3) Since , we know that the point is a local
maximum; since
f ′(3) = 0 and f ′′(3) > 0, the point ( 3, f (3) ) is a local minimum. Thus,
is a local maximum and (3, −3) a local minimum.(4) is unnecessary for this problem.
(5) f ′′(x) = 0 when x = , and f ′′ changes from negative to positive as x
increases through , so the graph of f has an inflection point. See Figure
N4–4.