f(x)=[sqrt((x-68)^2 + x^3-116x^2-417x+267460)] - 10

To find the minimum value of we need to check the value at the following three points (in increasing order). (You will need to use a numerical method, like Newton-Raphson to find one of these points.)

x1=? x2=? x3=?

Does this question tie in somehow with the question you had posted in

http://www.jiskha.com/display.cgi?id=1258707548 ?

The second equation is identical.

BTW, did you ever take a look at the solution I gave you then?
Sometimes it is frustrating to have spent quite a bit of time solving a problem, and then not even knowing if the student that posted the question even looks at it.

To find the minimum value of the function f(x), we need to check the value at three points in increasing order. One of the points can be found using a numerical method like Newton-Raphson.

Newton-Raphson is an iterative method to find the root of a function. In our case, since we want to find a point where the derivative of the function f(x) is zero, we can use Newton-Raphson to find that point.

To apply Newton-Raphson, we need to find the derivative of f(x) with respect to x. Let's do that:

f(x) = sqrt((x-68)^2 + x^3 - 116x^2 - 417x + 267460) - 10

Taking the derivative of f(x) with respect to x:

f'(x) = [1/2 * (x-68)^2 + x^3 - 116x^2 - 417x + 267460]^(-1/2) * (2(x-68) + 3x^2 - 232x - 417)

Now, we can use Newton-Raphson to find one of the points where f'(x) = 0. Iteratively, we can update x using the formula:

x_new = x - f(x) / f'(x)

We can start with an initial guess for x, and then use the above formula to get closer to the root. Let's call this point x1.

Once we have found x1 using Newton-Raphson, we can then find the other two points x2 and x3 by selecting two additional points in increasing order near x1. These points can be obtained by incrementing or decrementing x1 by a small value.

So, to find the three points:
1. Use Newton-Raphson to find x1 by iteratively applying the formula x_new = x - f(x) / f'(x) with an initial guess for x.
2. Choose two additional points x2 and x3 in increasing order near x1.

It's important to note that the exact values for x1, x2, and x3 cannot be determined without solving the equation numerically or using a more robust root-finding algorithm.