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) = 2x⁴ + 8x³ + 12x² - x - 2

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 3X minus 4 X cubed, plus 6 X2 + 5 X + 1 is concave up or concave down and identify any inflection points. We have 4 possible choices as our answers. For choice A, we have concave up on the interval from minus infinity to infinity and no inflection points. For choice B, we have concave up on the interval from minus infinity to 1, union with the interval from 1 to infinity, and concave down on the interval from -1 to 1, and then inflection point at X equal to 1. For choice C, we have concave down on the interval from minus infinity to infinity and no inflection points. And for choice D, we have concave up on the interval from minus infinity to -2, union with the interval from -2 to infinity, and an inflection point at X equal to -2. Now we're asked to determine the intervals on which our function is concave up or concave down, as well as identifying any inflection points. So what we need to do is take two derivatives with respect to X of our function G of X. So taking the first derivative, we'll have G of X. is equal to the derivative with respect to x of the quantity of 3x minus 4 X cubed. Plus 6 X squared. Plus 5 X Plus one. So taking this derivative, this is going to be equal to 12 x cubed. Minus 12 X squared. Plus 12 X Plus 5. Taking one more derivative, we'll get G of X. That's gonna be equal to the derivative with respect to X of the quantity of 12 X cubed minus 12 X squad. Plus 12 X Plus 5. And taking this derivative, this is going to be equal to. 36 X squad, minus 24 X. Plus 12. So, we need to find the critical points, so we'll set our second derivative here equal to 0 and solve for X. So we'll have 36 X squad minus 24 X. Plus 12 is equal to 0. So, I can divide both sides of this equation by 12, and I will get 3 X2 minus 2 X plus 1 is equal to 0. So now I need to solve this equation for X. And I noticed here that I have a quadratic equation, so I can use our quadratic formula to solve for X. So we call for the quantum drag formula that we're going to have X is equal to minus -2. Plus or minus the square root of the quantity of minus 2 squared, minus 4 multiplied by 1, multiplied by 3. In quantity, and that whole quantity is divided by the quantity of 2 multiplied by 3. So simplifying this expression, this is going to be equal to 2, plus or minus the square root of the quantity of 4 minus 12 in quantity, and the whole quantity is divided by 6. Simplifying further, this is going to be equal to 2, plus or minus the square root of -8, and that whole quantity is divided by 6. So, at this point, we noticed that we have a negative um term underneath the radical. And therefore, we will have no real solutions for X. And so since I have no real solutions, I have no critical points, and therefore I have no inflection points. So, one part of our answer is that we have no inflection points. So that whatever the concavity of our function is, it's going to be the same concavity over the entire real line. So, now we just need to determine the concavity on the interval, going from minus infinity. To infinity And here we just need to determine whether our second derivative is positive or negative. So we can choose any point in this interval to see whether our function is going to be positive or negative. So we'll choose X equal to 0, so we'll have G of 0. It's going to be equal to. 36 multiplied by 0 squared. Minus 24, multiplied by 0, plus 12, and evaluating this expression, this is going to be equal to 12, which is a positive quantity since it's greater than 0. And so that means that our function is going to be concave up on this interval. So that is the concavity for our function. So now we have both our concavity and the inflection points identified. We can look at our answer choices, and we see that the correct answer choice corresponds to answer choice A. We're concave up on the interval going from minus infinity to infinity, and there are no inflection points. 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.f(x) = 2x⁴ + 8x³ + 12x² - x - 2

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

f(x) = 2x⁴ + 8x³ + 12x² - x - 2