Inflection Points and Constants

Question

Hi, I'm stuck on this problem, could someone please walk me through it?

Let f(x) = ax³ -bx², where a and b are constants.

A) Find all possible critical points of f in terms of a and b.

B) Find all possible inflection points in terms of a and b.

C) If a = 1 and b = 6, what is the maximum, minimum, and inflection point of the function? Report your solutions as coordinate points.

Answer 1

a) critical points are when f(x) is at a local minimum or local maximum. to find those we need to set the derivative of f(x)=0

take the derivative of f(x) keeping a and b as constants and set it equal to zero:

f'(x) = 3ax²-2bx = 0

factoring:

x*(3ax-2b) = 0

solving for x to make the equation true:

so x = 0 and x = 2b/3a are critical points

b) inflection points can be found by setting the second derivative equal to zero

take the derivative of f'(x) keeping a and b as constants and set equal to zero:

f''(x) = 6ax-2b = 0

solving for x to make the equation ture:

x = b/3a so that is your inflection point

c) plug in values where a = 1, b=6

local max @ x = 0

f(0) = 0 so the local max is @ (0,0)

local min @ x = 2b/3a = 4

f(4) = -32 so local minimum is @ (4,-32)

inflection point @ x = b/3a = 2

f(2) = -16 so inflection point is @ (2,-16)