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.

g(t) = 3t⁵ - 30t⁴ + 80t³ + 100

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says determine the intervals on which the function G of X, which is equal to 2 x 6, minus 18 x 5th, plus 45 x 4th plus 50 is concave up or concave down, and identify any inflection points. We're given 4 possible choices as our answers. For choice A, we have concave up on the interval from minus infinity to 0 and concave down on the interval from 0 to infinity, and inflection point at X equal to 0. For choice B, we have concave up on the interval from minus infinity to infinity, and no inflection points. For choice C, we have concave up on the interval from minus infinity to 3, and concave down on the interval from 3 to infinity. And we have an inflection point at X equal to 3. And for choice D we have concave up on the interval from minus infinity to 0, union with the interval from 3 to infinity, and we have concave down on the interval from 0 to 3, and we have an inflection point at X equal to 0 and X equal to 3. Now we asked to determine the intervals for which our function is concave up or concave down, as well as identify any inflection points. So to determine to determine whether the function is concave up or concave down, we need to take two derivatives with respect to X of our function G of X. So taking the first derivative will have G of X. equal to the derivative with respect to X. Of the quantity of 2 X minus 18 x 5. Plus 45 X to the 4th. Plus 50 And so taking this derivative, this is going to be equal to 12 x 5. Minus 90 4th. Plus 180 execute. Plus 0. So we now need to take another derivative, so we'll have G of X, and this is going to be equal to the derivative with respect to x of the quantity of 12 x 5 minus 90 x 4th plus 180 x cubed. And taking this derivative, this is going to be equal to. 60 X to the 4th. Minus 360 X cubed. Plus 540. X squared Now we can actually factor out a 60X2d out of each of those terms, but this is going to be equal to 60X2. Multiplied by the quantity of X2 minus 6 X. Plus 9 in quantity. And we can actually factor that second term, that this is going to be equal to 60 X squad, multiplied by the quantity of X minus 3 in quantity squared. Now we need to find the critical points for this function. So we'll set this second derivative equal to 0. In doing so, we will have 60 X2, multiplied by the quantity of X minus 3, in quantity squared is equal to 0. So the solutions to this um equation is going to be X equal to 0. Since we can set our first term 60 X2 equal to 0, so that means that X equal to 0, and our second solution is going to be X equal to 3. Since the quantity of X minus 3 and quantity squared must be equal to 0, so that means that X has to be equal to 3. So these are our two critical points, and we need to determine um the concavity of our function around our critical points to see if we have any inflection points. So, we're gonna be looking at different intervals. So, the first interval that we're going to look at is going to be the interval, going from minus infinity. 20. And so what we can do to see whether our function is concave up or concave down is to choose a point in this interval and see whether our second derivative is positive or negative. So, we'll choose the point X equal to -1, and so we'll look at G of -1. And this is going to be equal to 60, multiplied by -1 squared multiplied by the quantity of -1 -3 in quantity squared. And so when we evaluate this expression, this is going to be equal to 960, which is a positive quantity since it's greater than 0. That means on this interval, our function is going to be concave up. So now we need to look at our next interval, which is going to be the interval going from 0 to 3. So again, we'll choose a point in this interval, so we'll choose X equal to 1, and determine the value of our second derivative at that point. We'll have G of 1, and that's going to be equal to. 60 multiplied by 1 squared, multiplied by the quantity of 1 minus 3 in quantity squared. So, evaluating this expression, this is going to be equal to. 240, which is going to be greater than 0. So again, our function is concave up on this interval. And this also means that our critical point at X equal to 0 is not an inflection point, since our concavity did not change. And so the final interval that we need to check is going to be the interval, going from 3 to infinity. So again, we'll choose a point, and here we'll choose X equal to 4. So we'll have G of 4 is equal to 60. Multiplied by 4 quad, multiplied by the quantity of 4 minus 3 in quantity squared. So, evaluating this expression, this is going to be equal to 960 again, which, as we found earlier, is a positive quantity, since it's greater than 0, and again, our function is concave up on this interval. So that means that X equal to 3 is not an inflection point. In fact, what we see here is that our function is going to be concave up. Over the entire interval, going from minus infinity. To infinity With no Inflection points. So this here is the answer that we were looking for. Now, if we compare our answer that we found to our answer choices. We see that the correct answer choice actually corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

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

g(t) = 3t⁵ - 30t⁴ + 80t³ + 100

Key Concepts

Concavity

Second Derivative Test

Inflection Points

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