# How do you find the x coordinates of all points of inflection, final all discontinuities, and find the open intervals of concavity for #f(x)=x^4-8x^3#?

points of inflection at

concave up for

concave down for

The second derivative is required for concavity and inflection points.

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To find the x-coordinates of points of inflection:

- Find the second derivative of the function.
- Set the second derivative equal to zero and solve for x.
- The x-values obtained are potential points of inflection.
- Use the second derivative test or examine the sign changes of the second derivative around these points to confirm points of inflection.

To find discontinuities:

- Identify any values of x for which the function is undefined.
- These values represent potential discontinuities.
- Check if the function approaches different values from the left and right side of these points.
- If there's a difference in the limit from the left and right side, it indicates a discontinuity.

To find the open intervals of concavity:

- Find the second derivative of the function.
- Determine where the second derivative is positive or negative.
- Positive second derivative indicates concavity upwards, negative second derivative indicates concavity downwards.
- Use test points within each interval to confirm the concavity.

For the function f(x) = x^4 - 8x^3:

- Find the first derivative: f'(x) = 4x^3 - 24x^2.
- Find the second derivative: f''(x) = 12x^2 - 48x.
- Set f''(x) = 0: 12x^2 - 48x = 0.
- Solve for x: x = 0 and x = 4.
- Use the second derivative test to confirm points of inflection.
- Test intervals between points of interest to determine concavity.
- Identify any discontinuities based on the function's domain.

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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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