Based on everything that we've learned so far, we know how to identify the global maximum and minimum values of a function by looking at its graph. We know that these are the absolute highest and lowest points that we see our function reach. But we won't always have the graph of a function to rely on and we'll still need to be able to find the global extrema just based on the function itself. And that's exactly what I'm going to show you how to do here, how to find the global maximum maximum and minimum of a function following a step-by-step process, starting first with the extreme value theorem. So let's go ahead and jump in here.
Now, before we can actually find the global extrema of a function, we first have to determine if that function even has a global max and global min. And that's what we can do using the extreme value theorem because we know that if we had a parabola that just went off to infinity, that function doesn't actually have a global maximum. So the extreme value theorem tells us that if a function f is continuous, meaning it doesn't have any breaks or holes in it just like we see in this function here, just a smooth continuous curve, and it is continuous on a closed interval from a to b, that means both a and b are included within that interval. Then if both of those things are true, then we know that our function f has a global maximum and a global minimum value within that interval from a to b. So really, we're checking for 2 things.
Is our function continuous, and is it defined over a closed interval? If those two things are true, we know that our function has a global maximum and global minimum value. So with our function f(x)=3x2+1, if we want to find if this function has a global maximum and minimum, we want to check those 2 things. Is this function continuous? Well, this is a basic polynomial, so I know that it is definitely continuous.
And is it defined over a closed interval? When we're asked specifically to look at the interval from negative 2 to 4, we can see that this interval is closed based on those square brackets, so this is also true. So, yes, this function f(x) does have a global maximum and global minimum value. Okay. Now that we know it has those values, how can we actually find them?
Once you verify that those values exist, you can then actually find your global maximum and minimum by testing the critical points and the endpoints of your function. These will then give you your global max and your global min. So let's actually work through this process step by step with our function f(x) here. The very first thing that we want to do is find the critical points of our function. Now in order to do that, we need to find that derivative and set it equal to 0 and also determine where it does not exist.
Now since we're working with this polynomial here, I want to find f'(x). Using the power rule, taking the derivative of 3x2+1 is just going to give me 6x. So to find my critical points here, I just need to set this equal to 0 because there is nowhere that this derivative does not exist. Now setting this equal to 0 gives me my single critical point x=0. So with step 1 done, we can move on to step 2 and now plug our critical points and our endpoints into our original function f(x).
One thing to note here is that you only want to plug critical points in that are inside of your closed interval. Now here, our closed interval is from negative 2 to 4, and 0 is within that interval. So I do want to proceed in plugging that in. So I want to plug 3 values in here, x=0, x=−2, and x=4. That's my critical point and my 2 endpoints.
I'm plugging these back into my original function f(x). So plugging 0 in here to that 3x2+1, that's going to give me 3×02+1, which is just equal to 1. Then plugging that negative 2 in, that's 3×(−2)2+1, which gives me a value of 13. And then finally, plugging 4 in, that's 3×42+1, which gives me a final answer here of 49. So I've plugged all of these values in.
We have completed step 2, and we can proceed to our final step where we're actually going to get our global max and global min. Now of the values that we just found in step 2, we want to look for our largest and our smallest values. Our largest value is going to give us our global maximum. Now looking at these three values, I have 1, 13, and 49. This 49 is clearly the largest value, meaning that my function has a global maximum of 49 at x=4.
Then, of course, our smallest value will then be our global minimum. Here, my smallest value I can see is 1, so that means that my function then has a global minimum of 1 at x=0. So in order to find the global extreme of a function, you first have to verify if you even have a global max and global min using the extreme value theorem. From there, you want to find your critical points. Then plug your critical points and endpoints back into your original function so that you can identify your global max and global min.
Feel free to let us know if you have any questions, and let's continue practicing in the next video.