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#?

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

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

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.

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.