determine all critical points for the function.\ny = 4x² - 128√x\na. x = 0, x = 4, and x = -4\nb. x = 0\nc…

Explanation:

Step1: Find the derivative of the function

The function is (y = 4x^{2}-128\sqrt{x}=4x^{2}-128x^{\frac{1}{2}}). Using the power rule ((x^{n})^\prime=nx^{n - 1}), we have (y^\prime=8x-\frac{128}{2}x^{-\frac{1}{2}}=8x-\frac{64}{\sqrt{x}}).

Step2: Set the derivative equal to zero

Set (y^\prime = 0), so (8x-\frac{64}{\sqrt{x}}=0). Multiply through by (\sqrt{x}) (since (x\geq0) for the original function (y = 4x^{2}-128\sqrt{x})) to get (8x^{\frac{3}{2}}-64 = 0). Then (8x^{\frac{3}{2}}=64), (x^{\frac{3}{2}} = 8), and (x=8^{\frac{2}{3}}=(2^{3})^{\frac{2}{3}}=4). Also, consider where the derivative is undefined. The derivative (y^\prime=8x-\frac{64}{\sqrt{x}}) is undefined at (x = 0) (since (\frac{1}{\sqrt{x}}) is undefined at (x = 0)).

Answer:

C. (x = 0) and (x = 4)