Extrema & points of inflection

Question

List the coordinates of where extrema or points of inflection occur then state whether the function is increasing or decreasing 

f(x)=x^3-4x^2-3x+4

Answer 1

f(x) = x³ - 4x² - 3x + 4

 

To find the extrema, compute the first derivative, f'(x), set it to zero and solve for x:

 

f'(x) = 3x² - 8x - 3

0 = 3x² - 8x - 3

0 = (3x+1)(x-3)

 

x = -1/3 and 3

 

To determine whether the extreme points are maxima or minima, take the second derivative, f"(x).  If it's positive at the extreme point, it's a minimum.  If it's negative, it's a maximum.  Test both extreme points.

 

To find inflection points, set the second derivative to zero and solve for x.

 

f(x) is increasing as it approaches a maximum or after a minimum.  It is decreasing as it approaches a minimum or after a maximum.  The intervals you need to check are: (-∞,-1/3), (-1/3,3), and (3.∞).