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.

h(t) = 2 + cos 2t on [0,π]

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says to analyze the concavity of the function H or T, which is equal to 1 minus cosine of 4T, and to find the intervals where it is concave up or concave down. Also identify any inflection points within the closed interval from 0 to pi divided by 2. We're given 4 possible choices as our answers. For choice A, we have concave up on the intervals from 0 to pi divided by 8, union with the interval from 3 pi divided by 8 to pi divided by 2, concave down on the interval from pi divided by 8 to 3 pi divided by 8, and we have inflection points that T is equal to pi divided by 8. And 3 divided by 8. For choice B, we have concave up on the interval from 0 to pi divided by 4. Union with the interval from divided by 2 to pi. We have concave down on the interval from pi divided by 4 to pi divided by 2, and we have inflection points that T is equal to pi divided by 4 and pi divided by 2. Your choice C, we have concave up on the interval from 0 to pi divided by 6. Union with the interval from pi divided by 3 to pi divided by 2. We have concave down on the interval from pi divided by 6 to pi divided by 3, and inflection points at T is equal to pi divided by 6, and pi divided by 3. And for choice D we have concave up on the interval from 0 to pi divided by 5. Union with the interval from 4 pi divided by 5 to pi. We have concave down on the interval from pi divided by 5 to 4 pi divided by 5, and we have inflection points that T is equal to pi divided by 5 and 4 pi divided by 5. Now we need to determine the intervals where our function HFT is concave up or concave down, as well as in the inflection points within the closed interval from 0 to pi divided by 2. So the first step to determining our concavity is we need to find the second derivative of our function, since that is what we use to determine concavity. So we need to take two derivatives of our function H of T with respect to T. So taking the first derivative, we'll have H of T. Which is equal to the derivative with respect to T of the quantity of 1 minus cosine of 4 T. So taking this derivative, this is going to be equal to the derivative of 1 with respect to T is 0. Then we'll have minus the derivative with respect to T of cosine of 4T, which is -4, multiplied by sine of 4 T. So simplifying this expression, this is going to be equal to 4, multiplied by the sign of 4 T. We need to take one more derivative that will have H of T, which is equal to the derivative with respect to T of the quantity of 4 multiplied by sine of 4 T. So taking this derivative, this is going to be equal to 16, multiplied by cosine of 4 T. So we'll now use our second derivative here to determine whether our function H of T is concave up or concave down. To recall that if our second derivative is positive over an interval, then our function is going to be concave up, and if our second derivative is negative over an interval, that means that our function is going to be concave down. And so here we're going to be looking over the interval, the close interval going from 0 to pi divided by 2. So we need to determine Whether our second derivative here, HT is positive or negative. So we need to look at the intervals in which our function cosine of 4T is positive or negative, since that is what determines the sign of our second derivative. So, cosine of 4 T. is positive, greater than 0. On the intervals from 0 to high divided by 8. As well as the interval of 3 pi divided by 8. 2 pi divided by 2. And so it's possible from both of these intervals, so we'll actually just union those two intervals. Now, cosine of 40. It is less than 0, it's negative. On the interval, going from pi divided by 8 to 3 pi divided by 8. So, what this means is that our function here. It's going to be concave up. When cosine of 4T is positive. And so in this case, our function HFT is going to be concave up. On the interval, from 0 to pi divided by 8. Union with the interval of 3 pi divided by 82 pi divided by 2. And our function is going to be concave down. When our second derivative is negative. Which is going to be on the interval from pi divided by 8 to 3 pi divided by 8. And so these are our intervals of concavity that we are looking for for this problem. So now we need to just determine our inflection points. And so, we first need to define our critical points for our second derivative, so we'll have our second derivative equal to 0, so we'll have 16. Multiplied by cosine of or T. is equal to 0. And we need to solve this or T. Now, our function here is going to be equal to 0. When cosine of 4 T is equal to 0, so that means that 40, Has to be equal to. Pi divided by 2 or 3 pi divided by 2. And here we're only considering these two values because T has to be within the interval from 0 to pi divided by 2, which means that 4T is going to be within the interval from 0 to 2 pi. So these are the only two values within that interval in which our codesine function is 0. So now we just need to solve for T, we'll divide both sides by 4, so we'll see that T is equal to pi divided by 8, and 3 pi divided by 8. And now I need to make sure that our concavity does change at those two points. And if I look at our intervals of concavity that we calculated earlier, we see that we do have a change concavity at pi divided by 8. It's going from concave up to concave down, and we also have a change at 3 pi divided by 8. Since we're going from concave down to concave up. So these two points are inflection points. So, now that I have the answer to our question, we can compare that to our answer choices, and we see that the correct answer choice corresponds to answer choice A. 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.

h(t) = 2 + cos 2t on [0,π]

Key Concepts

Concavity

Second Derivative Test

Inflection Points

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