Let ƒ(x) = (x - 3) (x + 3)²

d. Determine the ...

Question

Let ƒ(x) = (x - 3) (x + 3)²

d. Determine the intervals on which ƒ is concave up or concave down.

Accepted Answer

Welcome back, everyone. Analyze the function p of X equals x + 1 multiplied by X minus 62 to determine the intervals of concavity. So let's recall that whenever we are interested in the intervals of concavity, we want to identify the second derivative of the function, set it equal to 0 to identify any potential inflection points, or check if the second derivative doesn't exist. So in this case, if we're given PFX, we want to identify the second derivative, while in order to avoid the product rule, we can distribute the terms first. We are going to apply the full property, right, and simplify the factored form into a standard form. So PF X is equal to X + 1. Multiplied by x minus 62, that's what we're given. So what we have to do is simply square the second term. We're going to get X + 1 multiplied by x2 minus 2x multiplied by 6 gives us 12 x plus the square of the final term, right? So plus 36, here we are applying the square formula for the difference of two terms, and now we can foil, right? So we're going to get X. Imprints X2 minus 12 x plus 36 plus X2 minus 12 x plus 36. We basically have one in front, right? And now if we distribute, we're going to get X cubed minus 12 x 2 + 36 X plus X2 minus 12 x plus 36. And now combining the like terms, we begin with X cubed. Minus 12 x 2 plus X2, this gives us negative 11 X2d overall, plus 36 X minus 12 X. This gives us 24 X. And then plus 36. So, we have the standard form of PFX and now we can identify the second derivative. So, let's begin with the first one. The derivative of X cubed is 3x2d based on the power rule. We are subtracting the derivative of 11 X2, which is 22 X. And then we're adding the derivative of 24 X, which is 24. The derivative of 36 is 0. Now, the second derivative is the derivative of the previous result. We apply the power rule, the derivative of X where there's 2 X, so we get 6 X because there's 3. And we subtract 22 because the derivative of X is 1 and the derivative of 24 is 0. So now we set this to 0, it exists for all the domain because it's a linear function. And we want to identify a potential inflection point. So 6 X should be equal to 22, which gives us X equals. 22 divided by 6 or 11 divided by 3. So this is a potential inflection point, right? And we want to check if this function changes concavity. Before that point and after, so what we want to do is simply evaluate the sign of the derivative, specifically the second derivative, for any value that is less than 11 divided by 3. So let's say we take P at 0, right? Essentially, we want to show that we are evaluating the second derivative for X less than. 11/3 and we can choose any point that we want. So let's suppose that we take P at 0.0. And this gives us 6 multiplied by 0, which is 0 minus 22, that's -22. And that's less than 0, so it's concave down. Right, because whenever the second derivative is negative, the function is concave down. Now, He double prime. Let's evaluate it at a point that is greater than 11/3. So essentially we can take 12/3, which is 4, right? So X being greater than 11/3, let's suppose that we take P at 0.4, which gives us 6 multiplied by 4, 24 minus 22. This gives us 2, which is positive, and this is where the function is concave up. So indeed, 11/3 is the inflection point because the function changes concavity and we can say that it's concave up. When P is positive. So for any value greater than 11/3. So we're going to say that X belongs to the interval from 11/3 up to infinity, because any point greater than 11/3 satisfies the inequality. Then It is concave down. Elsewhere So when X belongs to the interval. From negative infinity up to 11/3. And that's it. And we have our final answers. Let's label them, and thank you for watching.

Let ƒ(x) = (x - 3) (x + 3)²

d. Determine the intervals on which ƒ is concave up or concave down.

Key Concepts

Second Derivative Test

Critical Points

Intervals of Concavity

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