Concavity Determine the intervals on which the following funct...

Question

Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.

f(x) = ³√(x - 4)

Accepted Answer

Below there, today we're gonna 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 following function is concave up or concave down. Identify any inflection points. K of X is equal to the cube root of X + 5. Awesome. So it appears for this particular problem we're asked to solve for three separate answers. We're trying to determine the intervals on which the following function of KFX is going to be concave up or concave down, and we're also asked to identify any inflection points. So those are our three separate answers that we're ultimately trying to solve for. Awesome. So with that said, our first step that we need to take in order to solve this problem is we need to determine the first derivative of the function K of X. So we're told that K of X is equal to there's a cube root of X + 5. So we're given this function, and when we take the first derivative of this function of K X, we will find that K of X is going to be equal to D by DX where D by DX is just a fancy way of saying we're taking the derivative with respect to X of X + 5 to the power of 1/3 means that's another way of writing the cube root. So we're taking the first derivative, so K where this prime symbol indicates the first derivative. So K of X is equal to D by DX of X + 5 to the power of 1/3, and we need to recall and use chain rule in order to perform this derivative. So when we do that, we will find that K of X is going to be equal to 1/3. Multiplied by X + 5 to the power of -23. Fantastic. So now our second step is we now need to determine the second derivative of this function of K X. So our second derivative, which we're trying to solve for now is K of X. So now that we're solving for K of X, we need to take K of X is equal to D by DX. Of 1/3 multiplied by X + 5 to the power of -2/3. Fantastic. So once again, you need to recall and use chain rule in order to perform this this differentiation. So when we do that, we will find that K is equal to D by DX of 1/3 multiplied by X plus, so it's 13 plus X + 5 with the power of -2/3, which is going to be equal to 1/3 multiplied by 2/3, multiplied by drum roll. X + 5, and then we need to take X + 5, and we need to take it to the power of -5/3. So now when we simplify, we will determine that K of X is going to be equal to -2 divided by 9, multiplied by X plus 5 to the power of 5/3 to the power of negative 5/3 to be precise. Awesome. So now, our next step now that we've determined the second derivative, what we need to do is we need to now switch our attention to determine the intervals of concavity. So we're trying to figure out the intervals of concavity now. So that's our next step. So we need to identify where K of X is greater than 0. Which that will mean it's concave up in that condition. So let's make a note of that. So if it's, so for the condition where K of X is greater than 0, it's gonna be concave up, and which I denoted as Cu. And if it's K of X is less than 0, that means it will be concave down. So I'm gonna denote that as CD. Fantastic. So we need to note that since K of X is equal to -2 divided by 9 multiplied by X + 5 to the power of -5/3, the sign of K X depends on X + 5 to the power of -5 divided by 3. So in order for X to be greater than -5, this X plus 5 to the power of -5 divided by 3 will be positive, making K of X negative, which is concave down. So let's make a note of that. So, X is greater than -5, so that will mean that In this case, it's going to be making our K. Concave down, so let's note that this will be CD for short, since it's concave down. And let us note that once again, that for the condition where X. is less than -5. We need to note that when X is less than -5, that will make that this value of X plus 5 to -5 divided by 3 will be negative. So it makes our value of K of X, which will mean that it's going to be concave up. Fantastic. So now our next step at this point is we now need to identify our inflection points. So we need to recall and note once again that inflection points occur when K of X is equal to 0. So let's make a note of that. So this is where K of X is equal to 0, or when K of X changes a sign. So make let's make a note of that. So changes sign. Fantastic. So now we need to note that K of X does not equal 0 for any value of X, but it changes the sign at X equals -5. So, therefore, X equals -5 will be an inflection point. So now we can summarize our results. So simply we can simply state that. Our summary of our answers that we found together, we can simply say, without a doubt that the function K of X will be concave up on the interval negative infinity -5, and we can say that the function K of X will be concave down on the interval -5, infinity. And once again, there will be an inflection point at X equals -5, and those are our final three answers. Hooray, we did it. We solved for all the answers that we needed to solve for. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.

f(x) = ³√(x - 4)

Key Concepts

Concavity

Second Derivative Test

Inflection Points

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