Increasing and decreasing functions. Find the intervals on whi...

Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = tan⁻¹ (x/(x²+2))

Accepted Answer

Below there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the intervals on which the function F of X is equal to inverse cosine of 2 X divided by X2 + 3 is increasing and decreasing. Awesome. So it appears for this particular problem we're asked to figure out or determine the intervals on which that this function of F of X is increasing and decreasing. So with that said, let's read off our multiple choice answers to see what our final answer might be. A is the function F of X is decreasing on parentheses square root of 3 square root of 3, and increasing on parentheses negative infinity square rooted 3 and square root of 3 infinity in parentheses. B is the function of F of X is increasing on parentheses square rooted 3 square root of 3, and decreasing on parentheses negative infinity square rooted 3 and square rooted 3 infinity. C is the function F of X is decreasing on infinity and increasing on negative infinity square rooted 3. And finally, D is the function F of X is increasing on parentheses negative square root of 3 infinity and decreasing on parentheses negative infinity square root of 3. OK. So first off, in order to determine the interval on which this function of F of X is equal to inverse cosine of, 2 X divided by X2 + 3 is increasing or decreasing, we must follow these steps. So first off, we need to for step one, find the derivative of F of X. So, in order to find the derivative of F of X, We must recall and apply the chain rule for the derivative of inverse cosine of G of X. So when we do that, we will find that F of X, where this prime symbol denotes the first derivative of F of X. So F of X is going to be equal to negative 1 divided by the square root of 1 minus G of X. Square. Multiplied by G of X, and let us note that G of X, in this case is going to be equal to 2 X divided by X2 + 3. Fantastic. So now we need to compute the value of G by recalling and using the quotient rule. So G of X is equal to X2 + 3, multiplied by 2 minus 2 X multiplied by 2 X, all divided by. X2 + 3, all the power 2, which is going to be all equal to when we write it in its most simple form, which I'm going to skip a few steps, but when we write this in its most simple form, we will find that we will have an answer for G of X is equal to negative 2x the power of 2 + 6 all divided by X2 + 3 to the power of 2. Fantastic. And once again, that is when it's in its most simplified form. So therefore, without a doubt, our derivative will become F of X is going to be equal to negative 1 divided by the square root of 1 minus 2 X divided by X to the power of 2 + 3, all to the power of 2. Multiplied by -2 x to the power of 2, plus 6 all divided by X2 + 3 to the power of 2, which when we simplify, we will find that this is equal to drum roll, 2 x 2 minus 6 all divided by. X to the power of 2 + 3, and we need to take X2 + 3, and we need to multiply the square root of X to the power of 4 + 2 X to the power of 2 plus 9. Fantastic. So, now our next step is we need to take. We need to make sure we note our F of X value, and now we need to take F of X and we need to set it equal to 0. So that's our second step. So our second step is we need to take F of X and we need to set it equal to 0. So we need to set it equal to 0 in order to find our critical points, which I denoted as CP. So in order to determine when F of X is equal to 0, we need to set our numerator, which we need to take our numerator, which is -2 X2 + 6 and set it equal to 0. And because the denominator is always positive, that will mean that X2 is equal to 3, which will mean that X is equal to plus or minus the square root of 3. So therefore, without a doubt, our critical points are going to be equal to square root of 3 and square root of 3. So X equals -2 root of 3 and square root of 3. Fantastic. So now, our third step is we need to determine the increasing and decreasing intervals. So when X is greater than the square root of 3, or when X is less than the negative square root of 3, so when X is less than square root of 3, we can go ahead. Write that 2 x 2 minus 6 is greater than 0. So that will mean as a result that F of X is going to be greater than 0. So that will mean that F of X is going to be increasing. On, so it's gonna be increasing on parentheses negative infinity square root of 3, and square root of 3 infinity, and for when negative square root of 3 is less than X and X is less than square root of 3, that will mean that 2 X to the power of 2 minus 6 is less than 0, so that will mean that F of X is going to be less than 0. So that will mean that F of X will be decreasing. On, so it's gonna be decreasing on negative square root of 3 square root of 3, and once again it's in parentheses. So therefore, without a doubt, when we go to look at our multiple choice answers once again, that will mean that our final answer has to be multiple choice answer letter A. Which states that it's gonna be increasing on parentheses negative infinity square rooted 3 and parentheses square rooted 3 infinity. And decreasing on parentheses negative square rooted 3 square root of 3. And once again, repeating it, following along the guidelines that how the multiple choice answer says, it has to be our final answer, without a doubt has to be multiple choice answer letter A, which states that the function F of X is decreasing on parenthesis negative square rooted 3 square rooted 3. And increasing on parentheses negative infinity square rooted 3, and square rooted 3 infinity in parentheses. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = tan⁻¹ (x/(x²+2))

Key Concepts

Derivative

Critical Points

First Derivative Test

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