1. graphing using the derivative

Basically i'm supposed to be using the derivative to find where the function is increasing and decreasing, the max and min values, and where the intervals of concavity and inflection points are. i understand how to get these (taking the first and second derivate) i'm just having a little trouble with the number line. Here's what i have so far. future apologies for having to writing in "calculator language"

A(x)= x((x+3)^(1/2)) (the (x+3)^(1/2) is the square root of X+3 )
A'(x)= (x/2 + (x+3)^(1/2))
(x+3)^(1/2)

Now i know that one of the critical points is -3 (because it makes the denominator undefined) I'm just not sure how to find the critical points on the numerator Any thoughts?

2. Originally Posted by mlckb1
Basically i'm supposed to be using the derivative to find where the function is increasing and decreasing, the max and min values, and where the intervals of concavity and inflection points are. i understand how to get these (taking the first and second derivate) i'm just having a little trouble with the number line. Here's what i have so far. future apologies for having to writing in "calculator language"

A(x)= x((x+3)^(1/2)) (the (x+3)^(1/2) is the square root of X+3 )
A'(x)= (x/2 + (x+3)^(1/2))
(x+3)^(1/2)

Now i know that one of the critical points is -3 (because it makes the denominator undefined) Actually, this would be an asymptote. Critical points are points where the derivative is equal to zero. Since the demominator can never be zero, the numerator must be equal to zero. I'm just not sure how to find the critical points on the numerator Any thoughts?
Set the numerator to zero, and solve for x.

Just set the numerator equal

3. Originally Posted by mlckb1
Basically i'm supposed to be using the derivative to find where the function is increasing and decreasing, the max and min values, and where the intervals of concavity and inflection points are. i understand how to get these (taking the first and second derivate) i'm just having a little trouble with the number line. Here's what i have so far. future apologies for having to writing in "calculator language"

A(x)= x((x+3)^(1/2)) (the (x+3)^(1/2) is the square root of X+3 )
A'(x)= (x/2 + (x+3)^(1/2))
(x+3)^(1/2)

Now i know that one of the critical points is -3 (because it makes the denominator undefined) I'm just not sure how to find the critical points on the numerator Any thoughts?
$A(x) = x\sqrt{x+3}$

$A'(x) = \frac{x}{2\sqrt{x+3}} + \sqrt{x+3}$

get a common denominator and put these two terms together ...

$A'(x) = \frac{x}{2\sqrt{x+3}} + \frac{2(x+3)}{2\sqrt{x+3}}$

$A'(x) = \frac{3x+6}{2\sqrt{x+3}}$

determine the critical values now.