110. Suppose the derivative of the function y = f(x) is
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Question

110. Suppose the derivative of the function y = f(x) is

y'=(x-1)^22(x-2)(x-4).

At what points, if any, does the graph of f have a local minimum, local maximum, or

point of inflection?

Accepted Answer

Hello. In this video, we are given the derivative of the function Y is equal to G of X. The derivative is defined as Y equal to the quantity X + 2 raised to the 30th power multiplied by X minus 3 multiplied by X + 1. We want to determine the points where the graph of G of X might have a local minimum or local maximum. So, in order to identify the locations of the local minimum or local maximum, the first thing we need to do is we need to identify the values of the possible critical points. In order to do this, we want to find the values of X that cause the derivative to be zero. So, in order to find the critical points, we will take our derivative X + 2, raised to the 30th power, multiplied by X minus 3, multiplied by X + 1, set it equal to 0, and solve for X. Since our function is in the form of a factor of X, we will go ahead and set each factor of X equal to 0. So we will have X + 2 raised to the 30th equal to 0, we have X minus 3 equal to 0, and we have X + 1 equal to 0. If we subtract one to both sides of the first equation, or to the 3rd equation, we have X's equal to -1. In the middle equation, adding 3 will give us X's equal to 3. Now, how do we solve the first equation? Well, we need to remove this X + 2 from the 30th power, and in order to do this, we are going to take the 30th root of both sides of this equation. That is going to simplify the equation to be X + 2 equal to 0, which will leave us with X's equal to -2. So we have three possible critical points. We have 3 critical points at -2, -1, and 3. But now what we need to do is we need to identify which of these are going to be a local maximum or local minimum. In order to do this, we are going to use the first derivative test. Let's go ahead and draw a number line. Now, on this number line, we are going to position our critical points. Again, we have -2, -1, and 3. And what we want to do is we want to plug in values around the neighborhoods of these points into the derivative and determine what the behaviors are based off the outputs. So what are the possible uh testing points that we can plug into the derivative? Well, these testing points are going to be -3. -1.5, 0, and 4. We will take these 4 points and plug them into the derivative. When we plug in -3 into the derivative, we get the value of 12. Since this is a positive value, this tells us that the graph is going to be increasing to the left of -2. Now, when we plug in -1.5 into the derivative, we get the output of 2.1. This tells us that the graph will be increasing between -2 and -1. When we plug in the value 0 into the derivative, we get the output, which is a very negative large number, a very large negative number. So what this means is that the graph will be decreasing between -1 and 3. And finally, if we plug in 4 into the derivative, we get the output of 1.1, which is positive, so the graph will be increasing from 3 to infinity. No We want to pay attention to the behaviors around these critical points. Which of the critical points have the behaviors switch when we pass through them? Well, if we take a look, if we pass through -2, the graph is positive but then continues increasing, so -2 is not a local maximum or minimum. When we pass through -1, the graph switches from increasing to decreasing. This behavior represents a local maximum value, so we have a local maximum. At the value X is equal to -1. And finally, when we pass through the value of 3, the graph switches from decreasing to increasing, which represents the behavior of a local minimum. So we have a local minimum at the value X is equal to 3, and this is going to be the solution to the problem. So I hope this video helps you in understanding how to approach this problem, and I will go ahead and see you all in the next video.

110. Suppose the derivative of the function y = f(x) is
y'=(x-1)^22(x-2)(x-4).
At what points, if any, does the graph of f have a local minimum, local maximum, or
point of inflection?

Key Concepts

Derivative and Critical Points

Second Derivative Test

Points of Inflection

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