3. For what value of k will x+k/x have a relative maximum at x = -2?

A. -4 B. -2 C. 2 D. 4 E. None of these

y = x + k/x

y' = 1 - k/x^2

1 - k/x^2 = 0
1 - k/4 = 0
k = 4

(D)

Ah, the quest for the relative maximum value! Let's put on our mathematical clown shoes and solve this problem, shall we?

First, let's find the derivative of the function x + k/x. This will help us identify the critical points and determine the value of k.

Using the quotient rule, we have:

d/dx (x + k/x) = (1)(x) - (x + k)(1/x^2) = (x - (x + k)/x^2).

To find the relative maximum, we need to set the derivative equal to zero and solve for x:

x - (x + k)/x^2 = 0.

Now, to make things easier, let's give x a little magic touch and multiply both sides of the equation by x^2:

x^3 - (x + k) = 0.

Expanding that equation, we get:

x^3 - x - k = 0.

Hmm, now we need to find the value of k that satisfies this equation when x = -2. We could go into a lengthy discussion about cubic equations and use some sophisticated clown tricks, but why don't we simplify this a bit?

When x = -2, the equation becomes:

(-2)^3 - (-2) - k = 0.

Simplifying that, we get:

-8 + 2 - k = 0.

Combining like terms, we have:

-6 - k = 0.

Now, let's solve for k:

k = -6.

Therefore, the value of k that will give x + k/x a relative maximum at x = -2 is none other than the mischievous -6! We've solved the mystery, my friend!

So, the answer is E. None of these.

To find the value of k that will make the expression x + k/x have a relative maximum at x = -2, we can follow these steps:

1. Calculate the first derivative of the expression:
f'(x) = 1 - k/x^2

2. Set the first derivative equal to zero and solve for x:
1 - k/x^2 = 0

3. Multiply both sides of the equation by x^2 to eliminate the fraction:
x^2 - k = 0

4. Rearrange the equation to isolate x^2:
x^2 = k

5. Find the value of x at x = -2:
(-2)^2 = 4

6. Substitute the value of x = 4 into the equation from step 4:
4 = k

Therefore, the value of k that will make x + k/x have a relative maximum at x = -2 is k = 4. So, the answer is D. 4.

To find the value of k that will make the function \(x + \frac{k}{x}\) have a relative maximum at \(x = -2\), we need to use calculus.

Here's how to do it:

1. First, differentiate the function \(x + \frac{k}{x}\) with respect to x to find the derivative:
- The derivative of x with respect to x is 1.
- The derivative of \(\frac{k}{x}\) with respect to x is \(-\frac{k}{x^2}\).

So, the derivative of the function is \(1 - \frac{k}{x^2}\).

2. Set the derivative equal to zero to find the critical points. We want to find the x-values where the slope is zero, which could correspond to a maximum, minimum, or an inflection point:
\(1 - \frac{k}{x^2} = 0\)

3. Solve the equation for x:
\(\frac{k}{x^2} = 1\)
\(k = x^2\)

4. Now we know that \(k = x^2\). Plug in the given value \(x = -2\) to find the corresponding value of k:
\(k = (-2)^2 = 4\)

Therefore, the value of k that will make \(x + \frac{k}{x}\) have a relative maximum at \(x = -2\) is 4.

So, the correct answer is D. 4.

You already answered the question, estupid