Identifying Extrema

In Exerc...

Question

Identifying Extrema

In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.

Accepted Answer

Welcome back, everyone. Find the X coordinates of the local minima and maxima of the continuous function F of X given the graph of its derivative F of X. Before we analyze the graph, let's recall how we can use derivatives to identify the local extreme points. First of all, if the derivative is positive. This means that the function is increasing. If the first derivative is negative. The function is decreasing. So basically, if we visualize a local maximum. The function must be increasing, so the first derivative should be positive, and then it should start to decrease after some point of interest X1, which is going to be a local maximum. So the derivative must become negative. We also have a local minimum. If the function is decreasing and then it starts to increase, right? Minimum at some point, let's call it X2. So first of all, we should have a derivative that is negative, and then passing that point, it should become positive. F of X should become positive. So we can analyze F of X, and specifically we're looking for a point where the derivative changes sign. So we have quadron 2. Where the derivative is positive, and then we can see that specifically at X equals 0. Our derivative changes a sign from being positive to negative, right? So from the right side, if we are approaching our curve, we can see that we have F of X approaches -5, from the left. F of X approaches 3, right? So from being positive, it becomes negative after it passes through X equals 0. What does that mean? Well, F of X. Changes from positive. Into negative, so the function was increasing, right? And then it started to decrease. And this indicates a local maximum. So we can say that local. Maximum I X equals 0. Now what else can we see? Well, when we consider quadron 4. We have a negative derivative. And then After X equals 3. We're moving to the first quadrant where the derivative is positive, right? So we had a negative derivative after X equals 3, that derivative becomes positive. So let's conclude that F of X. Changes from being negative into being positive. When we pass X equals 3. This indicates that the function was decreasing and then started to increase, and that's the characteristic of a minimum. So we have a local minimum. Add X equals 3. And those will be our final answers. Thank you for watching.

Identifying Extrema

In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.

Key Concepts

Derivative and Critical Points

First Derivative Test

Continuity of the Function

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