Help with finding first derivative and critical points?

[Brianna V]

Homework Statement

What are the critical points of function f(x) = 2 (x^2 + 4)^(1/2) - 4x + 24 ?

Homework Equations

When f'(x) equals 0 or is undefined, x is a critical number.

The Attempt at a Solution

The original function is f(x) = 2 (x^2 + 4)^(1/2) - 4x + 24 .
I got the derivative as f'(x) = [2x / (x^2 + 4)^(1/2)] - 4 .
What are the critical points? x = 0 is the only critical point I figure since 2x = 0, x = 0 (setting the numerator equal to zero). Any confirmation here?

Or do I need to put everything under a common denominator and figure stuff out that way?
i.e. 2x/[(x^2+4)^(1/2)] - 4[(x^2+4)^(1/2)]/[(x^2+4)^(1/2)]
= 2x - 4 /[(x^2+4)^(1/2)]
= 2x - 4 /[(x^2+4)^(1/2)] * [(x^2+4)^(1/2)]/[(x^2+4)^(1/2)]
= 2x[(x^2+4)^(1/2)]-4(x^2+4) /x^2+4
= 2x[(x^2+4)^(1/2)]-4x^2-16 /x^2+4
...But now what :S...

 

[Ray Vickson]

Homework Statement

What are the critical points of function f(x) = 2 (x^2 + 4)^(1/2) - 4x + 24 ?


((This is the actual original problem I am working with...
A man is on a bank of a river that is 2 miles wide. He wants to travel to a town on the opposite bank, but 6 miles upstream. He intends to row on a straight line not some point P on the opposite bank, and then walk the remaining distance along the shore. How far up the river should the point P be located to reach his destination in the least time if he walks 4 mph and rows 2 mph?))

Homework Equations

When f'(x) equals 0 or is undefined, x is a critical number.

The Attempt at a Solution

The original function is f(x) = 2 (x^2 + 4)^(1/2) - 4x + 24 .
I got the derivative as f'(x) = [2x / (x^2 + 4)^(1/2)] - 4 .
What are the critical points? x = 0 is the only critical point I figure since 2x = 0, x = 0 (setting the numerator equal to zero). Any confirmation here?

Or do I need to put everything under a common denominator and figure stuff out that way?
i.e. 2x/[(x^2+4)^(1/2)] - 4[(x^2+4)^(1/2)]/[(x^2+4)^(1/2)]
= 2x - 4 /[(x^2+4)^(1/2)]
= 2x - 4 /[(x^2+4)^(1/2)] * [(x^2+4)^(1/2)]/[(x^2+4)^(1/2)]
= 2x[(x^2+4)^(1/2)]-4(x^2+4) /x^2+4
= 2x[(x^2+4)^(1/2)]-4x^2-16 /x^2+4
...But now what :S...

The function has no critical points: it is strictly decreasing on ℝ. You can see this by writing f'(x) with a common denominator. You tried this, but you wrote it incorrectly on your second line above: you wrote
[tex]2x - \frac{4}{(x^2+4)^{1/2}},[/tex]
but possibly you meant
[tex] \frac{2x-4}{(x^2+4)^{1/2}},[/tex]
which would still be wrong. I suggest you start again, and be careful.

RGV