Question

Find the equation of the tangent lines at all inflection points of the graph of

f(x)=x^4-6x^3+12x^3-8x+3.

Answer 1

The inflection points occur where the concavity of the graph changes

First, we will find the inflection points

f'(x) = 4x³ + 18x² - 8

f''(x)= 12x² +36x

Put f''(x)=0

x= 0,-3

These are the inflection points.

For x=0,

f(0) = 3

f'(0) = -8

Slope at (0,3) is -8

For x=-3

f(3)= 81+6(-27)+24+3 = -54

f'(3) = 4(-27)+ 18(9)-8= 46

Slope at (3,-54) is 46

The equation of tangent at point (a,b) is given by

y-b = slope (x-a)

The equation of tangent at (0,3) is

y-3 = -8(x)

y+8x=3

The equation of tangent at (3,-54) is

y+54=46(x-3)

y-46x=-192