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.