Find the absolute maximum and minimum values of each function over the indicated interval, and i

Find the absolute maximum and minimum values of each...

Key Concepts

Endpoint Evaluation

When analyzing functions on a closed interval, it is crucial to evaluate the function's value at the endpoints. Since the absolute maximum or minimum may occur at these boundaries, comparing these values to those at the critical points is necessary for a complete analysis.

Derivative Calculation

Calculating the derivative of a function is a fundamental step in finding critical points. The derivative provides insights into the function's behavior and its rate of change, which are key in finding points where the function may reach extremum values.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval, then it must attain both an absolute maximum and an absolute minimum value within that interval. This theorem is fundamental in guaranteeing that the search for extrema is not in vain when working on closed intervals.

Critical Points

Critical points are values in the domain of a function where its derivative is either zero or undefined. These points are where the function could potentially have local maximums or minimums and are essential to investigate when determining absolute extrema on a given interval.

Transcript

00:01 Here we have the function f of x is equal to x plus 4 over x, and we closed interval from negative 8 to negative 1.

00:08 So here first we go ahead and differentiate and find the derivative.

00:12 That would be f prime of x.

00:14 We differentiate term by term, and we get, well, derivative over x is just 1.

00:18 And the derivative of 4 over x, well, this is 4 times x of negative 1.

00:24 So we get negative 4 times x to negative 2, which is minus 4.

00:30 4 over x squared.

00:33 Okay, so this is our derivative, 1 minus 4 over x squared.

00:37 And now from the critical values, we take our derivative and we set it equal to 0.

00:41 So we have 1 minus 4 over x squared equal to 0.

00:47 Well, that would imply that just x squared minus 4 is equal to 0, which implies that x squared is equal to 4 means that x is equal to positive or negative, so we have two critical values here, positive or negative 2.

01:06 So then, well, we list out all critical values and endpoints.

01:10 But notice here that positive 2 is not in our interval, right? so therefore, we can throw out positive 2 and say the only critical value here is negative 2.

01:23 So then we have our endpoints, negative 8, and then we have a critical value of negative 2 and our other end point of negative 1.

01:32 And then we go ahead and evaluate our given function, f of x, at those values...

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