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

Answer 1

There is no such #k#.

Let #f(x) = x+k/x#
the #f'(x) = 1-k/x^2#
In order for #f# to have #2# as a critical number we must have #k=4#.
However, #1-4/x^2# is negative on #(0,2)# and positive on #(2,oo)#, so #f# has a relative MINIMUM at #x=2#.
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Answer 2

There is a relative minimum when #x=2# with #k=4#, but there is no possible value of k which gives a relative maximum when #x=2#

Let

#f(x) = x+k/x#

Then differentiating wrt #x# we get

# f'(x) = 1 - k/x^2 #

And differentiating again wrt #x# we get:

# f''(x) = (2k)/x^3 #

the #f'(2) = 1-k/4#

At a maximum or minimum we require #f'(x) =0#, so for a maximum when #x=2# we must have #f'(2)=0#

# f'(2)=0 => 1 - k/2^2=0#
# :. 1-k/4 = 0#
# :. k = 4#

So When #k=4 => f'(x)=0# when #x=2# giving a single critical point

Now let's find the nature of this critical point. With #k=4# and #x=2#

# f''(2) = ((2)(4))/2^3 > 0 #, Hence this a relative minimum

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Answer 3

To find the value of (k) for which (x + \frac{k}{x}) has a relative maximum at (x = 2), we first find the derivative of the function and set it equal to zero to locate critical points. Then, we test the nature of these critical points to determine the value of (k) that yields a relative maximum at (x = 2).

Given the function (f(x) = x + \frac{k}{x}), its derivative is (f'(x) = 1 - \frac{k}{x^2}).

Setting (f'(x)) equal to zero, we have (1 - \frac{k}{x^2} = 0).

Solving for (x), we get (x^2 = k).

Since we want a relative maximum at (x = 2), we substitute (x = 2) into (x^2 = k), yielding (k = 4).

So, the value of (k) for which (x + \frac{k}{x}) has a relative maximum at (x = 2) is (k = 4).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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