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) = -x⁴ - 2x³ + 12x²

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says to analyze the concavity of the function and identify the inflection points. And we're given the function H of X is equal to -2 x 4th plus 8 X cubed minus 10 X squad. Now, we need to analyze the concavity of our function as the first part of this problem. So, to do so, we'll need to take two derivatives of our function H of X. So taking one derivative will have H of X, that's going to be equal to the derivative with respect to X. Of the quantity of minus 24th, plus 8 X cubed. Minus 10 x squared. So taking this derivative, this is going to be equal to. For all three terms, we're going to use our constant multiple and power rule. And so, finding the derivative for our first term, we're going to have -8, multiplied by X cubed. For the second term, we'll have plus 24 X squared, and then for the last term, we'll have minus 20 X. And we really need our second derivative here, so we'll take one more derivative, so we'll have H of X is equal to the derivative with respect to X. Of the quantity of minus 8 X cubed. Plus 24 X squad minus 20 X. So taking this derivative, this is going to be equal to, and again, here we, we will use our constant multiple and power rules for derivatives, and we will get minus 24 X squad. Plus 48 X minus 20. So now we need to actually identify possible inflection points, and to do that, we're going to set our second derivative here equal to 0 and solve for X. We will have minus 24 X squared. Plus 48 X. -20 is equal to 0. Now, we can simplify this equation by dividing both sides of the equation by -4, and this will give us 6 X squad. Minus 12 X. Plus 5 is equal to 0. And so here we have a quadratic equation, so we can solve this equation by using our quadratic formula. So recall for the quadratic formula, we will have X. This equal to -12. Plus or minus the square root of the quantity of -12 squared, minus 4 multiplied by 6. Multiplied by 5. And that whole quantity is divided by The quantity of 2 multiplied by 6. So simplifying this expression, this is going to be equal to 12. Plus or minus the square root of the quantity of 144 minus 120, and that whole quantity is divided by 12. Simplifying further, this is going to be equal to one. Plus or minus? The square root of 6 divided by 6, since we'll have square root of 24 divided by 12, we can factor out underneath the square root, a factor of the square root of 4, which will give me 2, and so 12 divided by 2 is 6, and what I'm left with underneath the square root sign is going to be a 6. So, this here is our possible locations of inflection points. So now we need to determine the concavity around these two points. The words start off by looking at the interval, going from minus infinity. 21 minus square root of 6 divided by 6. And so on this interval, we need to determine whether our second derivative is positive or negative. So we'll choose a point in this interval, and we'll choose X equal to 0. So, we'll evaluate H of 0, which is going to be equal to. Minus 24, multiplied by 0 squared. Plus 48 multiplied by 0. -20. And when we evaluate this expression, this is going to be equal to -20. Which is less than 0. And so since our second derivative here is negative, that means that this interval, our function is going to be concave down. Now we're gonna need to look at the next interval, which is going to be the interval from 1 minus the square root of 6 divided by 621 plus the square root of 6 divided by 6. And so in this interval, we'll choose a point to evaluate to determine the sign of our second derivative. So here we'll choose X equal to 1. So we'll have H of 1, that's going to be equal to -24, multiplied by 1 squared, plus 48 multiplied by 1 minus 20. And when we evaluate this expression, this is going to be equal to 4. And so since our second derivative here is actually greater than 0, so it's positive, so that means that on this interval, our function is going to be concave up. And then the final interval that we need to look at is going to be the interval. From 1 plus the square root of 6 divided by 6, to infinity. And again, we'll just choose a point on this interval, so we'll choose uh X is equal to 2. So we'll have H2 of 2 is equal to minus 24, multiplied by 2 squared. Plus 48, multiplied by 2 minus 20. And when we evaluate this expression, this is going to be equal to -20. Which, again, is less than 0, that means over this interval, our function is going to be concave down. So here we can see that the concavity of our function did change at X equal to 1 minus the square root divided by 6, and it also changed at the point of 1 plus the square root of 6 divided by 6. So those two points were indeed inflection points. So for a final answer, we'll see that our function is. Concave up. On the open interval From 1 minus the square root of 6 divided by 62 1 plus the square root of 6 divided by 6. Our function is concave. Down On the open interval from minus infinity to 1 minus the square root of 6 divided by 6. Union with the open interval. From 1 plus the square root of 6 divided by 6, to infinity. And then finally, we have inflection points. At the two values that we found for X. Which are X is equal to 1 minus the square root of 6 divided by 6, and X is equal to 1 plus the square root of 6 divided by 6. So this here is our final answer for this problem. 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) = -x⁴ - 2x³ + 12x²

Key Concepts

Concavity

Second Derivative Test

Inflection Points

Watch next