Find the maximum, minimum, and inflection points for the following function ? y = #(x-1)^4(x+2)^3#

Answer 1

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Answer 2

To find the maximum, minimum, and inflection points for the function (y = (x-1)^4(x+2)^3):

  1. Maximum and Minimum Points:
    • Take the derivative of the function and find critical points by setting it equal to zero.
    • Then, use the second derivative test to determine whether these critical points are maxima, minima, or points of inflection.
  2. Inflection Points:
    • Find the second derivative of the function.
    • Set the second derivative equal to zero and solve for (x) to find possible inflection points.
    • Determine the concavity of the function around these points using the second derivative test.

Let's proceed with the calculations:

  1. First Derivative: (y' = 4(x-1)^3(x+2)^3 + 3(x-1)^4(x+2)^2)

  2. Critical Points: Set (y' = 0) and solve for (x).

  3. Second Derivative: (y'' = 12(x-1)^2(x+2)^3 + 12(x-1)^3(x+2)^2 + 6(x-1)^4(x+2))

  4. Possible Inflection Points: Set (y'' = 0) and solve for (x).

  5. Evaluate Second Derivative Around Possible Inflection Points: Determine the concavity around these points using the second derivative test.

  6. Final Analysis:

    • Identify which critical points correspond to maximum, minimum, or points of inflection.
    • Provide the coordinates of these points as the final answer.
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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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