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

Below

As a result, the concavity changes, creating an inflexion point at x=4.

Consequently, the concavity changes, resulting in an inflexion point at x=2.

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Point of inflection is at

The graph is not scaled.

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

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

- Find the second derivative of the function ( f(x) ).
- Set the second derivative equal to zero and solve for ( x ) to find the potential points of inflection.
- 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.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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