Question

(1) the graph of every cubic polynomial has precisely one point of inflection

(2) f''(2)=0 then the graph of f must have a point of inflection at x=2

(3) If f ' (c)>0 then f is concave upward at x=c

Answer 1

anyone?!

Answer 2

(1) TRUE. the second derivative of a cubic polynomial is a linear polynomial, which has exactly one root, implying exactly one point of inflection.

Answer 3

(2) generally FALSE. the first derivative locates the relative maximum or minimum, or a horizontal tangent line.
special case when it is true. f(x) = (x-2)^n for integers n=3,4,5,...

Answer 4

(3) FALSE. the first derivative desribes whether the graph is rising (f'(c) > 0) or falling (f'(c) < 0)