Concavity of parabolas Consider the general parabola described...

Question

Concavity of parabolas Consider the general parabola described by the function f(x) = ax² + bx + c. For what values of a, b, and c is f concave up? For what values of a, b, and c is f concave down?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Analyze the concavity of the quadratic function HFX is equal to MX2 + NX plus P. Determine the conditions on M, N, and P for HFX to be concave up and concave down. Awesome. So it appears for this particular problem we're asked to solve for two separate answers. We're trying to determine the conditions on MNNP for HF X to be concave up, our first answer set, and then concave down our second answer set. So, that means our first answer we're trying to figure out the conditions for it to be concave up, and then our second answer is to determine the conditions for a concave down. So with that said, let's read off our multiple choice answers to see what our final answer might be. A is HF X is concave up if M is greater than 0 and concave down if M is less than 0, regardless of N and P. B is H of X is concave up if N is greater than 0, and concave down if N is less than 0, regardless of M and P. C is H of X is concave up if P is greater than 0 and concave down if P is less than 0, regardless of M and N. And finally, D is H of X is concave up if M is less than 0 and concave down if M is greater than 0, regardless of N and P. OK. So, first off, in order to solve this problem, we need to identify the second derivative of the function H of X is equal to MX2 + N X + P. So we need to note that the first derivative, which is H X, and H of X, when we take the first derivative of our original function for H X, you will find that the first derivative will yield us to multiplied by M, multiplied by X plus N. And the second derivative, which is denoted as H of X, and our second derivative, which means we're taking the derivative of H X. So when we take the derivative of H X, we will find that H of X will be equal to 2 M. So now, our next step, so that was our first step. Our second step now is we need to determine the condition for concavity. So we need to note that for HF X to be concave. Up, and I'll use an arrow pointing up. So for in order for H X to be concave up, the second derivative must be positive, so that means that H of X must be greater than 0. And for H X to be concave down, that will mean that Our second derivative, which will be H of X has to be less than 0. So, with that said, So we need to now state the conditions based on our second derivative that we determined together. So that will mean, without a doubt that H of X will be concave. Up. So once again, we're using our second derivative that we found together to see if it meets the following criteria. So we can state now, without a doubt that H of X is gonna be concave up if 2 multiplied by M is greater than 0, which will simplify to M is greater than 0. So that is our first answer. Hooray, we did it. Now we need to solve for the concave down condition, which I'm gonna just denote it as C for concave. So H of X will be concave down. If 2 multiplied by M is less than 0, which will once again simplify 2, M is less than 0, so that means that we found our second answer. Hooray, we did it. So, therefore, in conclusion, the function H of X is equal to MX2 + N X plus P will be concave up when M is greater than 0, and our function H of X. Will be concave down when M is less than 0, and this corresponds with multiple choice answer letter A, which states once again that H of X is concave up if M is greater than 0 and concave down if M is less than 0, regardless of NNP. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Concavity of parabolas Consider the general parabola described by the function f(x) = ax² + bx + c. For what values of a, b, and c is f concave up? For what values of a, b, and c is f concave down?

Key Concepts

Concavity

Second Derivative Test

Quadratic Function

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