Absolute maxima and minima Determine the location and value of the absolute extreme values of f

step 1 Let's find the absolute max and mince for this function. F of x equals x to the third minus 3x s

Absolute maxima and minima Determine the location and value...

Key Concepts

Endpoint Evaluation

Evaluating the function at the endpoints of a closed interval is a critical step in optimization problems. Even though critical points provide candidates for local extrema, the absolute maximum or minimum might be attained at the boundaries of the interval. By comparing the function’s values at both the endpoints and the interior critical points, one ensures that the true extreme values are identified.

Extreme Value Theorem

The Extreme Value Theorem guarantees that a function that is continuous on a closed interval will have both an absolute maximum and an absolute minimum on that interval. This fundamental concept assures us that when dealing with optimization problems in such a setting, we can always find extreme values by checking the endpoints as well as any interior points where the function's derivative is zero or undefined.

Critical Points

A critical point occurs where the derivative of the function is zero or does not exist. These points are significant in optimization because they are the likely locations of local maximum or minimum values. In the process of finding absolute extremes, critical points are evaluated alongside the endpoints of the interval to ensure all potential candidates for the maximum and minimum are considered.

Transcript

00:01 Let's find the absolute max and mince for this function.

00:03 F of x equals x to the third minus 3x squared on the interval of negative 1, 2, 3.

00:10 Now to do this, we have to do two things.

00:13 We have to find the critical numbers, and we have to evaluate the critical numbers and the end points into the original function.

00:19 So let's start by finding the critical numbers here.

00:25 And to do that, we see where the derivative is equal to 0.

00:29 So let's find the derivative first.

00:30 So f prime of x is going to be 3x squared minus 6x.

00:37 And we want to know when is this equal to 0.

00:42 So 3x squared minus 6x equals 0.

00:47 I'm going to factor out x.

00:48 So this is 3x minus 6 is equal to 0.

00:53 So right away we're going to get x being zero as one value.

00:58 And then when 3x minus 6 equals 0, okay, that's going to be when x equals 2.

01:07 You bring the 6 over, you divide by 3, you'll get x equals 2.

01:12 And so those are the two critical values for us.

01:15 And now we're going to do the second part, which is evaluate the critical numbers and the end points.

01:31 Okay, so let's first see what? the critical numbers are.

01:40 Okay, now one thing to always just always be aware of, because i think some people tend to struggle with this, you're evaluating it in this function, meaning you're going to take these critical numbers, 0 and 2, and you're going to plug them in for x here, not in the derivative.

01:57 So, f at 0 is going to be 0 cubed minus 3 times 0 squared, all right, well, it's just zero...

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