How do you find the local extrema for #y=sqrtx/(x-5)#?

First find the derivative and set the derivative equal to 0:

Using the quotient rule.

We now need to find the roots of this equation:

graph{sqrt(x)/(x-5) [-10, 10, -5, 5]}

As a result of the above and a quick look at the graph we can see that there are no turning points and as such there are no local extrema.

To find the local extrema for ( y = \frac{\sqrt{x}}{x-5} ), follow these steps:

1. Find the first derivative of the function ( y ) with respect to ( x ).
2. Set the first derivative equal to zero and solve for ( x ) to find critical points.
3. Determine the second derivative of the function.
4. Evaluate the second derivative at each critical point.
5. If the second derivative is positive at a critical point, it corresponds to a local minimum. If the second derivative is negative, it corresponds to a local maximum. If the second derivative is zero, the test is inconclusive.

Let's go through these steps:

1. First, find the derivative of ( y ) with respect to ( x ): [ y = \frac{\sqrt{x}}{x-5} ] [ y' = \frac{d}{dx}\left(\frac{\sqrt{x}}{x-5}\right) ]

2. Set ( y' ) equal to zero and solve for ( x ) to find critical points.

3. Find the second derivative of ( y ): [ y'' = \frac{d^2}{dx^2}\left(\frac{\sqrt{x}}{x-5}\right) ]

4. Evaluate ( y'' ) at each critical point to determine the nature of the extrema.

By following these steps, you can determine the local extrema of the given function ( y = \frac{\sqrt{x}}{x-5} ).