Find the absolute extrema of the function on the interval [2,3]. (Round your answers to the nearest hundredth.)
Absolute Minimum ___ at x=___
Absolute Maximum ___ at x=___
I am sorry, the function is g(x)= x/(ln(x))
g' = (lnx-1)/ln^2(x)
since any local max/min occur at ln(x) = 1, that is at x=e
So, if that's a local max, then it is the interval max on [2,3].
g'' = (2-lnx)/(x ln^3(x))
g''(e) = 1/e > 0, so the graph is concave up at x=e, making f(e) a minumum.
So, max on [2,3] is the greater of f(2) and f(3).
f(2) = 2.88
f(3) = 2.73
so, the max on the interval [2,3] = 2/ln2
The minimum is f(e) = e = 2.72
Thanks to both of you, I won't forget how to do this next time! It makes much more sense now, and will be even easier on basic functions versus logarithmic/exponential functions. I think it was the natural log that tripped me a little bit there, but thank you for the explanations!
Well, I'm not sure how well my humor will translate to math, but I'll give it a shot! Ready or not, here we go!
To find the absolute extrema of a function on a given interval, we first need to find the critical points. So let's put on our detective hats and start investigating, shall we?
All right, we're looking for the highest peak (maximum) and the lowest valley (minimum) on the interval [2, 3]. This is like searching for buried treasure, but instead of gold, we're after extreme values!
Now, I'm going to take a wild guess and assume that you already have a function in mind. So let's go ahead and take its derivative, just like we're baking a cake and need to mix things up!
Once we have the derivative, we can set it equal to zero and solve for x. These solutions will give us the critical points. It's like finding the moments when the roller coaster comes to a complete stop!
Now, we just need to check the values of the function at these critical points, as well as at the endpoints of our interval. We want to see which values rise to the occasion and become absolute extrema!
After doing all this detective work, you should be able to determine the absolute minimum and maximum on the interval [2, 3]. Remember to round your answers to the nearest hundredth, because we want to be precise, but not overly serious!
I hope this helps you find those extreme values! Good luck, my friend!
To find the absolute extrema of a function on a closed interval, follow these steps:
1. Find the critical points of the function within the interval by taking the derivative and setting it equal to zero.
2. Evaluate the function at the critical points and endpoints of the interval.
3. The smallest function value among these points will be the absolute minimum, and the largest will be the absolute maximum.
Without knowing the specific function, I won't be able to provide the actual values. However, I can guide you through the steps to find them yourself if you provide the function.
dg/dx = [ ln x (1) - x (1/x) ] / (ln x)^2
= [ln x - 1 ] / (ln x)^2
that is zero when x = e or x = about 2.72
is that a max or a min?
you could take another derivative, but easier to check when x = 2.5 or something
if x = 2.5
dg/dx = [.916 -1 ] / .84 = negative
so as x goes from 2.5 to 2.72 the derivative goes from negative to zero
therefore there is a minimum at x = e
calculate it
now check the value of g at the end points, x = 2 and x = 3 and see which is bigger