How do you find the x coordinates of all points of inflection, final all discontinuities, and find the open intervals of concavity for #y=2x^2+4x+4#?

Answer 1

The graph is an upward opening parabola.

It has no points of inflection because it is concave up everywhere.

(If you want to look at the second derivative, it is #y''=4# which is positive everywhere.)

Since it is a polynomial, discontinuities are absent from it.

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Answer 2

To find the x-coordinates of points of inflection:

  1. Find the second derivative: ( y'' = 4 ).
  2. Set ( y'' = 0 ) to find points of inflection: There are none for this function.

To find discontinuities:

  1. Solve the denominator of any fraction in the function to find discontinuities: There are none for this function.

To find open intervals of concavity:

  1. Find the second derivative: ( y'' = 4 ).
  2. Since ( y'' > 0 ) for all ( x ), the function is concave up everywhere.
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Answer 3

To find the x-coordinates of all points of inflection, discontinuities, and open intervals of concavity for the function ( y = 2x^2 + 4x + 4 ):

  1. Points of Inflection:
    Calculate the second derivative of the function, ( y'' ). ( y'' = 4 ). Set ( y'' = 0 ) and solve for ( x ). Since the second derivative is constant and never equals zero, there are no points of inflection.

  2. Discontinuities:
    There are no discontinuities in a polynomial function like ( y = 2x^2 + 4x + 4 ).

  3. Concavity Intervals:
    To determine the concavity of the function, examine the sign of the second derivative. Since ( y'' = 4 > 0 ) for all ( x ), the function is concave up for all real numbers.

Therefore, there are no points of inflection, no discontinuities, and the function is concave up for all real numbers.

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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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