How do you find the maximum, minimum and inflection points and concavity for the function #y" = 6x - 12#?

Answer 1

This is a straight line.

It's a line with no concavity or inflection points.

Range is #(-oo , oo)#
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Answer 2

To find the maximum, minimum, and inflection points, as well as determine the concavity of the function y'' = 6x - 12, we need to follow these steps:

  1. Take the second derivative of the given function to find the concavity: y'' = 6.

  2. Since the second derivative is a constant (6), it indicates that the function is concave up everywhere.

  3. To find the maximum or minimum points, we set the first derivative equal to zero and solve for x: y' = 3x^2 - 12x = 0 Factoring out 3x, we get: 3x(x - 4) = 0 This yields two critical points: x = 0 and x = 4.

  4. To determine whether these critical points correspond to a maximum, minimum, or neither, we use the second derivative test:

    • For x = 0: Plug into the second derivative (y'' = 6x - 12): y''(0) = 6(0) - 12 = -12. Since the result is negative, it indicates a maximum point at x = 0.
    • For x = 4: Similarly, y''(4) = 6(4) - 12 = 12 - 12 = 0. The second derivative test is inconclusive, so we need additional information.
  5. To find inflection points, we set the second derivative equal to zero and solve for x: 6x - 12 = 0 Solving for x, we get: x = 2.

  6. To confirm whether x = 4 is an inflection point, we check the concavity on either side:

    • For x < 2, y'' < 0 (concave down).
    • For x > 2, y'' > 0 (concave up). Thus, x = 2 is an inflection point.

To summarize:

  • The function has a maximum point at (0, -12).
  • It has an inflection point at (2, f(2)).
  • Since the function is concave up everywhere, there are no minimum points.
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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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