SOLUTION: Let g(x) = x |x + 5| Find all critical numbers (if any) of g. Give answers in increasing order. Enter DNE in any unused space(s). x = x = Please help I dont underst

Question 996088: Let g(x) = x |x + 5|
Find all critical numbers (if any) of g. Give answers in increasing order. Enter DNE in any unused space(s).
x =

x =
Please help I dont understand the whole set up of absolute value. I had someone help solve and got.
g(x) = x|x+5|
|x+5| = (x+5) for x ≥ -5
|x+5| = -(x+5) for x < -5
I don't understand this at all. Why is it ≥ or <
How is this solved..
So confused

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Rule: if y = |x|, then dy/dx = |x|/x

Apply the derivative
g(x) = x*|x+5|
g ' (x) = |x+5| + x*(|x+5|)/(x+5) ... product rule (also use the rule above)
g ' (x) = |x+5| * [1 + x/(x+5)]
g ' (x) = |x+5| * [1*(x+5)/(x+5) + x/(x+5)]
g ' (x) = |x+5| * [(x+5)/(x+5) + x/(x+5)]
g ' (x) = |x+5| * [(x+5+x)/(x+5)]
g ' (x) = |x+5|*(2x+5)/(x+5)

Solve g ' (x) = 0 for x to find the critical values
g ' (x) = |x+5|*(2x+5)/(x+5)
0 = |x+5|*(2x+5)/(x+5)
|x+5| = 0 or (2x+5)/(x+5) = 0
|x+5| = 0 or 2x+5 = 0

If |x+5| = 0, then x+5 = 0 and x = -5
If 2x+5 = 0, then x = -5/2 = -2.5

The critical numbers are -5 and -2.5
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