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)(x + 2)(x − 3)

Accepted Answer

Hello. In this video, we are going to be determining the points where the function H has local maxima or minima given by H X equal to X minus 15, multiplied by X + 21, multiplied by X minus 13. So, just to quickly rewrite the derivative given to us, we have H X equal to X minus 15, multiplied by X + 21, multiplied by X minus 13. Now, in order to identify the local maximum or minimum, we need to first find the critical points of the problem. In order to find the critical points, we want to find the values of X that cause the derivative to be zero. So we are going to set the derivative equal to 0 and solve for X. By using the zero property, we can set each factor of X equal to 0, and that is going to give us X minus 15 equal to 0, X plus 21 equal to 0, and X minus 13 equal to 0. And if we solve for X in each of these equations, we get X's equal to 1315, and negative 21, which are going to be our critical points. Now, what we are going to do is we are going to identify the local maximum or minimum by using the second derivative test. But in order to use the second derivative test, we have to take the derivative of the first derivative of H of X. Now, since we have a product of 3 terms, or 3 factors of X, we are going to use the product rule, but we are going to set one of the functions as a product of 2 terms. So we will call this our G of X, and we will call X minus 15 or F of X for the product rule. No, The product rule is defined as the derivative of F multiplied by G, and this will be F multiplied by G plus G multiplied by F. In this case here, we labeled our F as X minus 15. The derivative of this X function is just going to be one. Now for our G function, we label G to be X + 21, multiplied by X minus 13. And in order to take the derivative of this function, we will also have to use the product rule. That is going to be the first function X + 21, multiplied by the derivative of the second function, which is 1, plus the second function X minus 13, multiplied by the derivative of the first function, which is just 1. This will leave us with X + 21 plus X minus 13. And if we simplify by combining like terms, we get 2 X minus, plus. 2 eggs plus 8. This is going to be our G for the overall product rule. Now that we have the 4 pieces that we need for the product rule. If we plug this into the derivative, we get H of X is equal to F of X, which is X minus 15, multiplied by the derivative of G of X, which is 2 X + 8. Plus, our original G of X, which is X + 21, multiplied by X minus 13. Multiplied by the derivative of F of X, which is going to be one. Now what we are going to do is we are going to simplify this second derivative. By foiling X minus 15 and 2 X + 8, we get 2 X squared plus 8 X minus 30 X minus 120. And foiling X + 21 and X minus 13 gives us positive X squared minus 13 X plus 21 X minus 273. And finally, by combining like terms, H of X is equal to 3 X2 minus 15 X minus 393. So this is going to be the simplified form of our second derivative. Now that we've identified our critical points, and we have our second derivative, we will now identify the local maxima or minima by using the second derivative test. The second derivative test is defined as the following. If we take a value and plug it into our second derivative, and the second derivatives value is greater than 0, this implies that we have a minimum value. And if we take a value plugging into our second derivative, and it's less than 0, this implies that we have a maximum value. So what we are going to do is we are going to take our critical points, plug it into the second derivative, and determine the maximum and minimum values. So, H of 13 is going to give us the value of negative 81, which is less than 0. H of -21 will give us the value of 1,245, which is greater than 0, and H of 15 is equal to 57. Which is greater than 0. So, what does this tell us about the critical points you've identified? Well, this tells us that we have a local maximum. At the value, X is equal to 13, and we have a local minimum or local minimums at the values X is equal to 15 and 21. And these are going to be the solution to this given problem. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

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)(x + 2)(x − 3)

Key Concepts

Critical Points

First Derivative Test

Sign Analysis of Derivative

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