Questions: Graph (y=64 x-x^3) using the following. a. the window ([-10,10]) by ([-10,10]) b. a window that shows 2 turning points b. Choose the correct graph of the function using a window that shows two turning points. A. B. C. D.

Solution Steps

The given function is \( y = 64x - x^3 \). To find the turning points, we need to find the first derivative and set it to zero.

The first derivative of \( y = 64x - x^3 \) is: \[ y' = 64 - 3x^2 \]

Set \( y' = 0 \): \[ 64 - 3x^2 = 0 \] \[ 3x^2 = 64 \] \[ x^2 = \frac{64}{3} \] \[ x = \pm \sqrt{\frac{64}{3}} \] \[ x = \pm \frac{8}{\sqrt{3}} \] \[ x = \pm \frac{8\sqrt{3}}{3} \]

Substitute \( x = \pm \frac{8\sqrt{3}}{3} \) back into the original function to find the corresponding y-values: \[ y = 64\left(\frac{8\sqrt{3}}{3}\right) - \left(\frac{8\sqrt{3}}{3}\right)^3 \] \[ y = 64\left(\frac{8\sqrt{3}}{3}\right) - \left(\frac{512 \cdot 3\sqrt{3}}{27}\right) \] \[ y = \frac{512\sqrt{3}}{3} - \frac{512\sqrt{3}}{9} \] \[ y = \frac{512\sqrt{3}}{3} - \frac{512\sqrt{3}}{9} \] \[ y = \frac{512\sqrt{3}(3-1)}{9} \] \[ y = \frac{1024\sqrt{3}}{9} \]

The correct graph should show two turning points at \( x = \pm \frac{8\sqrt{3}}{3} \) with the corresponding y-values. The graph that fits this description is option C.

Final Answer