Question

the graph of ( y = x ^ { 2 } - \frac { 16 } { x } ) has a horizontal tangent at the point ( p ( a, b ) ). find the coordinates of ( p ). give your answer in the form ( ( m, n ) ), where you replace ( m ) and ( n ) with the coordinates you found.

Explanation:

Step1: Find the derivative

The derivative of \(y = x^{2}-\frac{16}{x}\) is \(y'=2x+\frac{16}{x^{2}}\).

Step2: Set the derivative equal to 0

Since the tangent is horizontal, \(y' = 0\). So \(2x+\frac{16}{x^{2}}=0\). Multiply through by \(x^{2}\) (assuming \(x
eq0\)) to get \(2x^{3}+16 = 0\), then \(x^{3}=-8\), and \(x=-2\).

Step3: Find the \(y\) - coordinate

Substitute \(x = - 2\) into \(y=x^{2}-\frac{16}{x}\). \(y=(-2)^{2}-\frac{16}{-2}=4 + 8=12\).

Answer:

\((-2,12)\)