Question

In Exercises \(13-20\) , find all points of inflection of the function. $$y=x^{3}(4-x)$$

Short answer

The points of inflection are \((0,0)\) and \((2,16)\).

Step-by-step solution

Find the first derivative of the function

Using standard derivative rules to find the derivative of \(y=x^{3}(4-x)\) we get \(y' = 3x^{2}(4-x) - x^{3}\).

Find the second derivative of the function

The next step is to find the second derivative of \(y\). Let's differentiate \(y' = 3x^{2}(4-x) - x^{3}\), and we get \(y'' = 6x(4-2x) - 3x^{2}\).

Set the second derivative to zero

For inflection points, we set the second derivative equal to zero and solve for \(x\). So, let's set \(y'' = 6x(4-2x) - 3x^{2} = 0\). Solving this equation gives us \(x = 0, 2\).

Verify the points of inflection

Plug these values into the original equation \(y = x^{3}(4-x)\) to get the coordinates of the points of inflection. Thus, the points of inflection are \((0,0)\) and \((2,16)\).