Let f(x) be a polynomial function such that f(4)= -1, f'(4)=2 and f"(4)=0. If x<4, then f"(x)<0 and if x>4, then f"(x)>0.

Question

The point (4,-1) is which of the following for the graph of f?

How do you figure out this problem. Which answer choice is right and why?

a) relative maximum

b) relative minimum

c) critical value

d) inflection point

e) none of the above

Answer 1

From the second derivative, f" = x - 4

f' = (1/2)x^2 - 4x + c1

f'(4) = 2 = (1/2)(4)^2 - 4(4) + c1

f' = (1/2)x^2 - 4x + 10

f = (1/6)x^3 - 2x^2 + 10x + c2

f(4) = -1 = (1/6)(4)^3 - 2(4)^2 + 10(4)  + c2

f = (1/6)x^3 - 2x^2 +10x - (59/3)

A cubic polynomial with no "hump" or "dip" in it.  Think of a different version of plain old x^3.

Zeroes of the first derivative occur at 4 ± 2i in complex plane.

The point (4, -1) is an inflection point, foil (d).