How do you find the points of Inflection of #f(x)=2x(x-4)^3#?

Question

Samantha Adkison · Accepted Answer

Below

#f(x)=2x(x-4)^3# #f'(x)=2xtimes3(x-4)^2+2(x-4)^3# #f'(x)=2(x-4)^2(3x+x-4)=2(x-4)^2(4x-4)# #f''(x)=2times(2(x-4)(4x-4)+(x-4)^2(4))# #f''(x)=2(8x^2-40x+32+4x^2-32x+64)# #f''(x)=2(12x^2-72x+96)# #f''(x)=2(12)(x^2-6x+8)# #f''(x)=24(x-4)(x-2)# For points of inflexion, #f''(x)=0# ie #24(x-4)(x-2)=0# #x=4,2# Test #x=4# At #x=3#, #f''(x)=-24# At #x=4#, #f''(x)=0# At #x=5#, #f''(x)=72# As a result, the concavity changes, creating an inflexion point at x=4. Test #x=2# At #x=1#, #f''(x)=72# At #x=2#, #f''(x)=0# At #x=3#, #f''(x)=-24# Consequently, the concavity changes, resulting in an inflexion point at x=2.

Elijah Abram · Answer

Point of inflection is at #x=4# or #x=8/3#

Points of Inflection appear where the curve changes from being concave to convex or vice versa. This happens when second derivative i.e. #(d^2f)/(dx^2)=0#

As #f(x)=2x(x-4)^3#

#(df)/(dx)=2(x-4)^3+2x(x-4)^2=2x^3-24x^2+96x-128+2x^3-16x^2+32x=4x^3-40x^2+128x-128#

and #(d^2f)/(dx^2)=12x^2-80x+128#

This is #0#, when #12x^2-80x+128=0#

or #3x^2-20x+32=0#

or #(x-4)(3x-8)=0#

Hence point of inflection is at #x=4# and #x=8/3#

The graph is not scaled.

diagram{2x(x-4)^3 [-10, 10, -70, 30]}

Luke Alvarez · Answer

undefined To find the points of inflection of $ f(x) = 2x(x-4)^3 $, you need to follow these steps:

1. Find the second derivative of the function $ f(x) $.
2. Set the second derivative equal to zero and solve for $ x $ to find the potential points of inflection.
3. Test these potential points of inflection using the second derivative test to confirm whether they are indeed points of inflection.

First, let's find the second derivative of $ f(x) $:

$ f'(x) = 2(x-4)^3 + 2x \cdot 3(x-4)^2 $
$ f''(x) = 6(x-4)^2 + 2 + 6x(x-4) $

Next, set the second derivative equal to zero and solve for $ x $:

$ 6(x-4)^2 + 2 + 6x(x-4) = 0 $

Solve for $ x $ from this equation.

After finding the potential points of inflection, use the second derivative test to determine whether each point is a point of inflection. If the second derivative changes sign at a point, then that point is a point of inflection.