Find all relative extrema of the function. `f(x)=x-6 sqrtx` , x>0this is what I've come up with 1-(6x^1/2)=0 -6x^1/2=-1 x^1/2=1/6 ... I have more but am not sure it is right. Thankyou for your...

Find all relative extrema of the function. `f(x)=x-6 sqrtx` , x>0

this is what I've come up with 1-(6x^1/2)=0

-6x^1/2=-1

x^1/2=1/6

...

I have more but am not sure it is right.

Thankyou for your help.

Asked on by kwin13

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lemjay | High School Teacher | (Level 3) Senior Educator

Posted on

`f(x)=x-6sqrtx`

To determine the relative extrema, determine f'(x).

`f'(x)= d/(dx) x - d/(dx)6sqrtx`

Let's take the derivative of each term.

For the first term,

>>> `d/(dx)x = 1`

For the second term, apply the power rule of derivative which is `d/(dx) (cx^n) = c*nx^(n-1)` .

>>> `d/(dx) 6sqrtx= d/(dx)6x^(1/2)=6*1/2x^(-1/2) = 3/x^(1/2)`

So, f'(x) is:

`f'(x) = 1-3/x^(1/2)`

Set f'(x) equal to zero. And solve for x.

`0=1-3/x^(1/2)`

`3/x^(1/2)=1`

`3=x^(1/2)`

To isolate x, square both sides.

`9=x`

Then, substitute value of x to f(x).

`f(x)=x-6sqrtx=9-6sqrt9=9-6*3=6-18=-9`

Hence, at `xgt0`, the relative extrema of `f(x)=x-6sqrtx` is `(9,-9)`.

Sources:

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