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(x − 1)

Accepted Answer

Welcome back everyone. Determine the points where the function H has local maxima, or minima given the H of X equals 3 X multiplied by 2 minus X. So, first of all, what we want to do is set the derivative equal to 0 to identify the critical points of the function. This means that we X multiplied by 2 minus X is equal to 0. We have two factors to consider. This means that either our X is equal to 0 or 2 minus X is equal to 0, right? 3 has no effect, we can simply divide the equation by 3. And we can clearly see that we only have 2 factors to consider. So, solving these equations, we can say that X is either equals to 0. There is no necessity to solve it because it already gives us our first solution. And for the second one, we can just add X to both sides to show that X equals 2 is our second critical point. From here we're going to plot a number line and add the critical points 0 and 2, and we're going to divide our number line into 3 intervals from negative infinity up to 0, from 0 to 2, and from 2 to infinity. And we can perform the first derivative test, which basically means that we want to test specific points within each interval. For the first interval, we need to choose any point that is less than 0, such as -1, and we want to check the sign of the first derivative, which gives us 3 multiplied by -1 multiplied by 2 minus -1. We get -3, multiplied by 2 minus -1, so 3, right? -3 multiplied by 3 gives us -9. We have shown that the derivative. In the first interval is negative, which essentially means that our function is decreasing, right? Our next goal is to evaluate each prime at any point between 0 and 2, such as 1, which gives us 3 multiplied by 1 multiplied by 2 minus 1, which is 3 multiplied by 1, that's 3. And 3 is positive, which essentially means that our function begins to increase, right, from 0 to 2. And then we want to valid each prime at any point above 2, such as 3, which gives us 3 multiplied by 3, multiplied by 2 minus 3, and that's -9. Which is less than 0, it's negative, and this tells us that the function again is decreasing from 2 up to infinity. So now if we look at our sketch, we can essentially conclude that we have a minimum at 0 because the function was decreasing and starts to increase afterwards, and then at 2, we have a maximum. The function was increasing before 2 and started to decrease after 2. So, let's conclude our findings. Local Minimum At X equals 0 and local. Maximum At X equals 2. Those will be 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:

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

f′(x) = x(x − 1)

Key Concepts

Critical Points

First Derivative Test

Local Maximum and Minimum

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