Analyzing Functions from Derivatives

Question

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

b. On what open intervals is f increasing or decreasing?

f′(x) = x(x − 1)

Accepted Answer

Welcome back, everyone. Determine the intervals on which the function H is increasing or decreasing given its derivative H X equals 3x multiplied by 2 minus X. First of all, what we have to do is identify the critical points of H2 X by setting its derivative equal to 0. So we're setting 3 X multiplied by 2 minus X equal to 0. We can ignore the 3, right? Essentially we can divide both sides of the equation by 3 and show that x multiplied by 2 minus X is equal to 0, which gives us two solutions. We can consider our first factor X, which can be equal to 0 to give us a product of 0 or 2 minus X, which we set equal to 0. So, our first solution is X equals 0, and for the second one, we can show that X is equal to 2, right? So those are the critical points. And what we want to do is simply add them on a number line. So we're going to add X equals 0. X equals 2, we're going to divide the number line into 3 intervals between negative infinity and 0, between 0 and 2, and between 2 and infinity. And what we want to do is simply check the sign of the derivative in each interval. We have to recall that if each prime. is positive. This means that the function H is increasing, right? And if each is negative, then our function is decreasing. So let's perform the first derivative test. Let's evaluate H at any point that is less than 0, such as H at -1, which gives us we multiplied by -1, multiplied by 2 minus negative, or 1. And this gives us -3 multiplied by 3, which is -9. It is negative, right, which essentially means that the function is decreasing before 0. Now, let's validate H at 1, which is between 0 and 2. Which gives us 3 multiplied by 1, multiplied by 2 minus 1, and that's equal to 3, which is positive. It tells us that the function is increasing in this interval, and then we can evaluate each prime at, let's say, 3, which is above 2, right? So we get 3. Multiplied by 3, multiplied by 2 minus 3, which is -9. And again, this is negative, which shows us that the function is decreasing. So the function is decreasing between negative infinity and 0, increasing between 0 and 2, and then decreasing again between 2 and infinity. So let's conclude our findings. Let's Add H. Of X is increasing, and then let's add the interval. When X belongs to the interval between 0 and 2, and then H of X is decreasing. When X belongs to the interval from negative infinity up to 0. Union meaning and. From 2 up to infinity, those will be our final answers to this problem. Thank you for watching.

Analyzing Functions from Derivatives

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

b. On what open intervals is f increasing or decreasing?

f′(x) = x(x − 1)

Key Concepts

Derivative and Critical Points

Increasing and Decreasing Intervals

Sign Analysis of Derivatives

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