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 − 1)²(x + 2)

Accepted Answer

Welcome back, everyone. Determine the intervals on which the function H is increasing or decreasing given its derivative H X equals X + 6 squad, multiplied by X minus 7. So first of all, we want to identify the critical points of the function, meaning we're setting the derivative equal to 0. X + 6 squad multiplied by X minus 7. This should be equal to 0 to give us two critical points, right? So either x + 6 is equal to 0, or X minus 7 is equal to 0. Solving each equation, we get X equals -6 for the first one we subtract 6 from both sides. For the second one, we can add 7 to both sides, which gives us X equals 7. And now what we can do is simply plot a number line. We can add the critical points. One of them is -6 and the other one is 7. And these two critical points divide the number line. Into 3 intervals. So what we want to do is simply perform the first derivative test and check the sign of the first derivative. We're going to begin with H at any point that is less than -6, such as -7, and we can substitute it into the function or basically the function of the derivative, right? So this gives us -7 + 6 squared multiplied by -7 minus 7. Now we have to recall that the square of a number is always positive. And now we're multiplying by -7 minus 7, which is -14. That number is negative, right? So when we multiply a positive number and a negative number, we get a negative number. The most important thing here is the sign itself. We know that for Any point that is less than -6, we get a negative derivative. We can then evaluate each prime at any point between 6 and 7, such as 0, which gives us 0 + 6 squared. Again, this is going to be positive because we're squaring a number multiplied by 0 minus 7. So, a positive number multiplied by -7, which is a negative number. Again, this gives us a negative number, right? So, we still have a negative number. Between 6 and 7 when it comes to the sign of the first derivative, and finally, we want to evaluate each prime at any point above 7, such as 8. This gives us 8 + 6 squared multiplied by 8 minus 7. So, a positive number because it's a square multiplied by one, which is positive. So, the product between a positive number and another positive number is going to be positive. Let's add a positive sign above 7, and now let's recall that a function is increasing if the derivative is positive, right? If our first derivative is negative. This means that the function is decreasing. So what can we tell? Well, essentially we can see that our function has a negative derivative from negative infinity up to 7. And then our derivative becomes positive after 7 up to infinity. And this allows us to conclude the results. We can say that H. Of X is increasing. If X belongs to the interval from 7 up to infinity. And then each of X is decreasing if x belongs to the interval from negative infinity. Up to 7. That's it. We have our final answers and 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 − 1)²(x + 2)

Key Concepts

Critical Points

Increasing and Decreasing Intervals

Sign Analysis

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