Find the critical numbers for each continuous function on the given closed, bounded interval, an

Find the critical numbers for each continuous function on...

Key Concepts

Extreme Value Theorem

The Extreme Value Theorem asserts that if a function is continuous on a closed and bounded interval, then it must have both an absolute maximum and an absolute minimum on that interval. This theorem underpins the process of checking both critical points and endpoints when identifying extrema.

Closed Interval Method

The closed interval method involves checking the values of a continuous function at its critical numbers as well as at the endpoints of the interval. Because the Extreme Value Theorem guarantees that a continuous function on a closed interval will attain its maximum and minimum, evaluating these points determines the absolute extrema.

Critical Numbers

Critical numbers are the values in the domain of a function where its derivative is zero or undefined. They are significant because they are possible candidates for local minima, local maxima, or points of inflection in the function's graph.

Derivative

The derivative of a function measures its instantaneous rate of change. Taking the derivative is a key step in finding critical numbers, as setting the derivative equal to zero helps identify where the function's slope is zero, indicating potential extrema.

Product Rule

When differentiating a function that is the product of two functions, the product rule is applied. This rule states that the derivative of a product f(t)g(t) is f'(t)g(t) + f(t)g'(t), and is essential when dealing with expressions where the variable appears in multiple factors.

Transcript

00:01 For this problem, we've been given the function, f of t equals t times e to the negative 2t.

00:06 And our goal is to find the critical numbers for this function.

00:11 So let's take a look at what a critical number is.

00:15 Critical numbers happen at one of two places.

00:17 It's either where the derivative of the function equals zero or it's undefined.

00:24 So if we've been tasked with finding the critical numbers, our first goal needs to be to go back to our original function, and first find the derivative.

00:33 Once we have the derivative, we'll see if either of these two cases occurs, and if so, what values of x make those conditions true.

00:42 So back to our function.

00:44 This is a product, t times e to the negative 2t.

00:48 So we'll need to use the product rule.

00:50 Product rule says we take the first function, t, times the derivative of the second.

00:55 Well, the derivative of an e function is just itself, in this case, e to the negative 2t times the derivative of that exponent, which in this case is negative 2.

01:06 So first function times the derivative of the second plus the second function times the derivative of the first, which is just one.

01:16 So that gives me my function.

01:18 Now, e to the power, to any power like they have here, e to the negative 2t, there is no value that i can put in for t that will make this undefined.

01:29 So that is not going to happen in this case.

01:32 But we need to look at this option...

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