37. What value of a makes f(x) = x^2 +(a/x) have
a. a lo...

Question

37. What value of a makes f(x) = x^2 +(a/x) have

a. a local minimum at x = 2?

b. a point of inflection at x = 1?

Accepted Answer

Hello there. Today we're going to 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. What value of C makes H of Z equal to 3 Z2 + C divided by Z have a point of inflection at Z equals -2. Awesome. So it appears our final answer that we're ultimately trying to solve for our end goal is we're trying to find a value of C, which makes this function of H Z equals 32 plus C divided by Z to have an inflection point at Z equals -2. So now that we know that we're ultimately trying to figure out what this value of C is to basically meet all the conditions set to us by the problem itself. Let's read off our multiple choice sensors to see what our final answer might be. A is 12, B is 3 halves, C is -24, and D is 24. Awesome. So first off, in order to solve for this problem, our first step is we need to compute the 1st and 2nd derivatives of H of Z. So let us begin by computing, or rather finding the 1st and 2nd derivatives of H of Z. So let us note that the first derivative, which will be Hz, that's what the prime symbol means. So H Z, so H once again Z is the first derivative of our function of H of Z. So, Let us know that we're trying to find H of Z, but let us also quickly note that we're also told originally that H of Z, our function, is originally equal to 3 Z2 plus C divided by Z. That's our original function for H of Z. So when we take our first derivative, we will find that H of Z is going to be simply equal to 6 Z minus C divided by Z2. Fantastic. Now we need to find our second derivative, so we need to find H of Z because once again we're trying to find the second derivative. So when we take our answer that we got for H Z, which once again is 6 Z minus C divided by Z2, so once again it's 6 Z minus C divided by Z2, and we take the derivative of our result from H Z, we will find that H Z, our second derivative, will be equal to 6 plus. 2 C divided by Z to the power of 3. Fantastic. So now our second step that we need to take is we need to recall and apply the condition for a point of inflection at, and once again, let us note that it's at the equals -2. So for a point of inflection at Z equals -2, the second derivative must be equal to 0 at the point, meaning it has to be at H-2 is equal to 0. So that means we need to substitute Z equals -2 into H Z. So when we do that, we will find that H of Z. Which once again we're plugging in a value of -2 and for Z, so H of -2 is going to be equal to 6 plus 2C divided by -2 to the power of 3, which will be equal to 6 + 2C divided by ne8, which will be equal to 6 minus C divided by 4. Fantastic. So now we need to set this expression equal to 0, so we need to take 6 minus C divided by 4 is equal to 0, which we do a little bit of moving around, we will find that negative C divided by 4 minus 6. So once again we're trying to rearrange our equation to isolate and solve for C all by itself, and when we do that using a little algebra, we will find that C is equal to 24. So before we declare this our final answer, we need to check to see if it's an inflection point. So, if it's not an inflection point, then that means that this valid, this answer will be invalid, meaning it's not correct. So, once again, our third step now is we need to check to see if this is an inflection point. So, in order to do that, as we should recall, we need to substitute our value for C equals 24 back into our original function, which is H of Z is equal to 3 Z 2 plus C divided by Z. So once again, we're trying to check that this is an inflection point, so we're taking our value of C equals 24, and we're substituting it back into our original function for H of Z. So, with that said, What we need to do is we need to take H of Z is equal to 32 to the power 2 + 24 divided by Z. So, when we check for H, Of Z, so H Z will be equal to 6 Z minus 24. Divided by Z2d. Fantastic. And then finally, we need to check for H of Z. So H of Z will be equal to 6 + 48 divided by, so we're gonna take 6 + 48. Divided by Z to the power of 3, fantastic. So now we need to confirm if Z equals -2 is indeed an inflection point, and check the sign of the second derivative around Z equals -2. So let us note at this stage that for, and let's make a note of this in red. So for Z is less than -2, choose a value like Z is equal to -3. So with that said, we could take H of -3 is equal to 6. Plus 48 divided by -3 to the power of 3, which will be equal to, let me plug that into our calculator and so it will be equal to 38 divided by 9 or 38 9, and let us note that this is a positive value, fantastic. So now we need to take for the condition for when Z is greater than -2, so we need to choose a value like Z is equal to -1. So when we do that, we will find that H of -1. is equal to 6 + 48 divided by -1 to the 3, which will be equal to, when we have plugged that into our calculator and so, it'll be equal to -42, which is a negative answer or a Negative value answer. So since the second derivative changes signed from negative to positive at Z equals -2, this indeed confirms, as we should recall, this will confirm that Z equals -2 is indeed an inflection point. So therefore, without a doubt, the value is C, that makes H of Z equal to 3 Z 2 plus C divided by Z, to have a point of inflection at Z equals -2 will have to be, once again, C is equal to 24, and that is indeed our final answer. Hooray, we did it. So looking back at our multiple choice answers, that will mean, without a doubt that our final answer has to be multiple choice answer letter D, which once again is 24. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

37. What value of a makes f(x) = x^2 +(a/x) have
a. a local minimum at x = 2?
b. a point of inflection at x = 1?

Key Concepts

Local Minimum

Point of Inflection

Derivative Calculation

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