How do you find all points of inflection given #y=2x^2+4x+4#?

Question

Cameron Alexander · Accepted Answer

There are no points of inflection.

Points of inflection are places where the concavity of a function changes. Analytically, this happens when the sign of the second derivative #f''(x)# of a function changes. We first find #f''(x)#: #f(x) = 2x^2 + 4x + 4 # #f'(x) = 4x + 4 # #f''(x) = 4# The second derivative is a constant value (4), meaning that the second derivative is defined always (it's 4), and it will never change sign. Thus, there are no points of inflection. This is demonstrated also by the graph, which is a "bowl up" parabola, and thus never changes from bring "bowl up" to "bowl down".

Axel Alvarez · Answer

undefined To find the points of inflection for the function $y = 2x^2 + 4x + 4$, follow these steps:

1. Calculate the second derivative of the function.
2. Set the second derivative equal to zero and solve for $x$.
3. Once you have the $x$-values where the second derivative equals zero, plug these values into the original function to find the corresponding $y$-values.

Let's go through these steps:

1. The first derivative of $y = 2x^2 + 4x + 4$ is $y' = 4x + 4$.
2. Taking the derivative of $y'$, the second derivative is $y'' = 4$.
3. Setting $y''$ equal to zero gives $4 = 0$, which is not solvable because it's a constant.
4. Since the second derivative is a constant, there are no points of inflection for the function $y = 2x^2 + 4x + 4$.

In summary, the function $y = 2x^2 + 4x + 4$ does not have any points of inflection because its second derivative is a constant, indicating no change in concavity.