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#?

Answer 1

points of inflection at #x=0# and #x=4#
concave up for #(-oo, 0)# and #(4, oo)#
concave down for #(0,4)#

The second derivative is required for concavity and inflection points.

#f'(x)=4x^3- 24x^2#
#f^('')(x) = 12 x^2 - 48x#
#= 12 x(x-4)#
#f^('')(x)= 0# at #x =0# and #x=4#
Make a sign chart. To the left of #x=0#, the second derivative is positive. To the right of #x=4#, the second derivative is positive. Between #0# and #4# the second derivative is negative.
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Answer 2

To find the x-coordinates of points of inflection:

  1. Find the second derivative of the function.
  2. Set the second derivative equal to zero and solve for x.
  3. The x-values obtained are potential points of inflection.
  4. 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:

  1. Identify any values of x for which the function is undefined.
  2. These values represent potential discontinuities.
  3. Check if the function approaches different values from the left and right side of these points.
  4. 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:

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

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

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

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.

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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