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

step 1 Let's find the absolute max in mince for this function. F of x equals sine 3x on the interval fr

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

Key Concepts

Differentiation

Differentiation is the process of computing the derivative of a function, which provides the rate at which the function is changing. By taking the derivative and setting it equal to zero, one can find critical points where the function's growth may switch direction, helping to locate potential extreme values. This analytical tool is fundamental in the study of calculus and optimization problems.

Endpoint Evaluation

In problems involving finding absolute extreme values on a closed interval, it is important to evaluate the function at the endpoints of the interval in addition to its critical points. Since the extreme values of a continuous function may occur at the boundaries of the interval, this step ensures that no potential maximum or minimum is overlooked.

Critical Points

Critical points are points in the domain of a function where the derivative is zero or does not exist. These points are essential in the process of finding extreme values because they are potential locations where the function's rate of change shifts direction, marking local maxima, minima, or saddle points, which can be candidates for absolute extreme values when compared against the endpoints.

Absolute Extreme Values

Absolute extreme values refer to the highest and lowest values a function attains over a particular interval. Identifying these values involves evaluating the function at all critical points and the endpoints of the interval, then comparing the results to determine the maximum and minimum values over the entire domain.

Extreme Value Theorem

The Extreme Value Theorem guarantees that if a function is continuous on a closed interval, then it must have both an absolute maximum and an absolute minimum on that interval. This theorem provides the foundational assurance that the search for extreme values is valid and that they indeed exist when working with continuous functions over closed and bounded intervals.

Transcript

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

00:04 F of x equals sine 3x on the interval from negative pi over 4 to pi over 3.

00:11 So to find absolute max in mins, we have to find the critical values where the derivative is equal to 0.

00:17 And then evaluate the critical values and the endpoints in the original function and then see which ones are the largest and smallest outputs.

00:25 So let's find the derivative first.

00:26 F of x so we have a nice chain rule here so uh derivative of sign is going to be cosine keep the inside and then the derivative of the inside of three of x is three so we're going to multiply this by three and so then we get this equals three cosine of three x now i want to know when is this equal to zero well that's only going to happen when the cosine part of it is equal to 0.

01:02 Now when is cosine equal to 0? well, if you just think of a regular cosine theta being 0, well that's when theta is pi over 2, negative pi over 2, 3 pi over 2, and so on.

01:30 And so what we want to know actually is when is 3x equal to these values.

01:40 Now, they can't, obviously, there's infinitely many of these, but because we have this boundary condition, we're only going to have so many.

01:48 And so we're actually having to check and find out.

01:50 So we'll start with the easiest one, pi over two.

01:53 So when is 3x equal to pi over two? well, that's going to be when x equals pi over six.

02:03 So there's the first one.

02:07 And what about when 3x equals 3 pi over 2? oh, well, okay, well then that's when x is going to be equal to pi over 2.

02:22 Oh, but now we have a problem.

02:24 Well, that's not a problem.

02:26 But notice how pi over 2 is bigger than pi over 3.

02:28 So it's actually out of the domain.

02:31 It's out of the boundary.