Analyzing Functions from Derivatives

Question

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

c. At what points, if any, does f assume local maximum or minimum values?

f′(x) = (x − 1)²(x + 2)

Accepted Answer

Welcome back, everyone. Determine the points where the function H has local maxima or minimal given H X equals 3 multiplied by X + 4 squad multiplied by X minus 2. A says local minimum at x equals 2. B local maximum at X equals 2. C local minimum at X equals -4, and the local maximum at X equals -4. So let's analyze the given derivative. We have H X equals 3 multiplied by x + 4 squad, multiplied by x minus 2. What we have to do is simply identify the critical points of the function, and this is done by setting the derivative equal to 0. When we set it equal to 0, we get 3 multiplied by X + 4 squad, multiplied by X minus 2 is equal to 0. So we have two factors. Either x + 4 is equal to 0, or X minus 2 is equal to 0, which gives us two possible solutions. X can be equal to -4 and X can be equal to 2. So now let's add these numbers to a number line. And before we do that, we have to understand that if we're looking for a local maximum or minima, we want to see where the derivative changes that sign. For example, if we have a positive derivative, our function is increasing. If we have a negative derivative, that means the function is decreasing. So it classifies our critical point XC as the local maximum, right? This is exactly what we want to do. We want to identify the change in sign. So to simplify the problem, we can recall that X + 4 squad is going to be greater than or equal to 0 for all Xes because it's a square of a number, right? So it has no effect. On the sign of our inequality. Meaning the only point that we want to consider is X equals 2. We don't need to consider X equals -4 because it has no effect on the sign of the derivative. So, let's consider the point X equals 2. We're going to add it on a number line, and from here what we're going to do is simply perform the first derivative test. Let's evaluate each prime at any point, which is less than 2, such as 0, which gives us we multiplied by 0 + 4 squad multiplied by 0 minus 2. Which gives us a positive number. We already know that 3 multiplied by 4 squad is positive and 0 minus 2 is negative. So positive multiplied by a negative gives us a negative number, right? It is negative. And now we want to evaluate H at any point above 2, such as 3, which is going to give us a positive number we multiplied by 0+. 3 squared multiplied by 3 minus 2, another positive number. So the product of 2 positive numbers is going to be positive. Now what have we shown? Well, we have shown that our derivative is negative. Before we reach 2, and it becomes positive when we pass 2, right? So, if our derivative was negative, this means that the function was decreasing. And then after X equals 2, the derivative is now positive, meaning the function starts to increase. This indicates a local minimum at x equals 2. Therefore, the correct answer to this problem corresponds to the answer choice A, local minimum at X equals 2. Thank you for watching.

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

c. At what points, if any, does f assume local maximum or minimum values?

f′(x) = (x − 1)²(x + 2)

Key Concepts

Critical Points

First Derivative Test

Behavior of Polynomial Functions

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