Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. \[ y=3 x^{4}+16 x^{3}=x^{3}(3 x+16) \]

Step 1 ：First, we find the derivative of the function \(y = 3x^4 + 16x^3\) to identify the critical points. The derivative is \(y' = 12x^3 + 48x^2\). The critical points are where the derivative is equal to zero or undefined, which are \(x = -4\) and \(x = 0\).

Step 2 ：Next, we find the second derivative of the function, which is \(y'' = 36x^2 + 96x\). The inflection points are where the second derivative changes sign, which are \(x = -8/3\) and \(x = 0\).

Step 3 ：We then evaluate the function at the critical points and at the endpoints of the domain (if any) to determine the absolute extreme points. The y values at the critical points are \(y(-4) = -256\) and \(y(0) = 0\).

Step 4 ：To determine whether the critical points are local maximums, local minimums, or neither, we examine the sign of the first derivative on either side of these points. The critical point \(x = -4\) is a local maximum because the sign of the first derivative changes from positive to negative at this point. The critical point \(x = 0\) is not a local maximum or minimum because the sign of the first derivative does not change at this point.

Step 5 ：To determine whether the inflection points are indeed points where the function changes concavity, we examine the sign of the second derivative on either side of these points. The inflection points are indeed \(x = -8/3\) and \(x = 0\) because the sign of the second derivative changes at these points.

Step 6 ：Finally, to determine the absolute extreme points, we compare the y values of the critical points and the endpoints of the domain. The highest y value is the absolute maximum, and the lowest y value is the absolute minimum. The absolute maximum is \(y = 0\) at \(x = 0\), and the absolute minimum is \(y = -256\) at \(x = -4\).

Step 7 ：\(\boxed{\text{Final Answer: The local maximum is at the point } (-4, -256)\text{. There are no local minimums. The inflection points are at } (-8/3, -4096/27) \text{ and } (0, 0)\text{. The absolute maximum is at the point } (0, 0)\text{, and the absolute minimum is at the point } (-4, -256)\text{.}}\)