William W. answered • 10/27/21
Experienced Tutor and Retired Engineer
Take the derivative and set it equal to zero to find the first critical points.
Secondly, to find other critical points, determine where the derivative DNE on the interval.
Thirdly, look at the end points of the interval.
Determine the function values of all these and compare them. One will be the absolute min, another the absolute max.
It's hard to determine exactly what your function is:
f(x) = 4[(x + 1)/x] or is it f(x) = 4[x + 1/x] so I'm not going to proceed further than just the explanation above.
Larry S.
so when determining the function values do I input the values into the original equation and not the derivative to figure out the absolute min and max?10/27/21
William W.
Yes10/27/21
Larry S.
and do I just use the closed interval values into the original function to get the min and max?10/27/21
William W.
Use all three [1) “derivative set equal to zero” critical points, 2) “derivative DNE” critical points (if there are any), and 3) “interval end points”]. The greatest function value of all 3 is the absolute max on the interval.10/27/21
Larry S.
I'm confused on how to find the derivative DNE critical points, my professor never taught me that10/27/21
William W.
So since f(x) = 4(x + 1/x) = 4x + 4/x = 4x + 4x^-1 then the derivative f'(x) = 4 - 4/x^2. This derivative is not defined for x = 0 (Note that the function itself is also not defined at x = 0.) This would be a derivative DNE critical point HOWEVER, since it is not in your interval, who cares? So, setting the derivative equal to zero and solving you get critical points of x = 1 and x = -1. The only one of those that is in your interval is x = 1 (just happens to be one of the endpoints of your interval as well. So, using the original function, plug in x = 1 and you get f(1) = 8. Then plug in x = 4 (your other interval endpoint and you get f(4) = 17. So the only possible min or max in this case (on the interval [1,4] is x=1 (with a function value of 8) and x = 4 (with a function value of 17). So the absolute min occurs at x = 1 and is 8 while the absolute max occurs at x = 4 and is 17.10/27/21
Larry S.
the equation was the second you listed, so f(x) = 4[x + 1/x]10/27/21