Calculus
posted by Salman .
f(x)=[sqrt((x68)^2 + x^3116x^2417x+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 NewtonRaphson to find one of these points.)
x1=? x2=? x3=?

To find the minimum of the function, you need to find the derivative, f'(x). At the local minimums, the derivative equals zero. That gives you two of the three points, and for which you can solve using the quadratic formula.
Since the function is not a polynomial (because of the squareroot function), its domain is not (∞,∞).
You will need to find the domain of the function, namely the interval of which the expression under the squareroot sign stays positive. The said expression is a cubic and crosses the xaxis only once, so it is easy to find the zero using NewtonRaphson method. In case you have difficulties, it is around 43. 
Additional note:
do not forget to test the first two critical points (where f'(x)=0) for maximum or minimum. 
what is the exact value of the 43 term? i cant find it

Tell me what values you've got so far.
It should be quite straight forward, because the slope is almost uniform in that stretch.
Perhaps you didn't get the right derivative, or formula. 
Oh, it's not a factoring process, the third point is to be solved using NewtonRaphson. The first two are the solutions of the quadratic.

I cant find the value of x1. im getting it around 43.2423, but this answer is not correct

Mine agrees with yours up to 43.24...
To how many figures are you supposed to calculte, and how many lives do you have left (assuming you have a computer exercise).