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#?
The graph is an upward opening parabola.
It has no points of inflection because it is concave up everywhere.
Since it is a polynomial, discontinuities are absent from it.
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To find the xcoordinates of points of inflection:
 Find the second derivative: ( y'' = 4 ).
 Set ( y'' = 0 ) to find points of inflection: There are none for this function.
To find discontinuities:
 Solve the denominator of any fraction in the function to find discontinuities: There are none for this function.
To find open intervals of concavity:
 Find the second derivative: ( y'' = 4 ).
 Since ( y'' > 0 ) for all ( x ), the function is concave up everywhere.
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To find the xcoordinates of all points of inflection, discontinuities, and open intervals of concavity for the function ( y = 2x^2 + 4x + 4 ):

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