Let a be a given real number such as a 2. Let the function f: R → R be defined by
f (x) = | x | • (a + x)
a) Find the zeropoint. Are there points where f is not differentiable?
b) Determine where f increases and decrease, and provide any local and global extremal
and its extreme values.
c) Determine where f is convex and concave, and provide any turning points.

Question

Let a be a given real number such as a> 2. Let the function f: R → R be defined by
f (x) = | x | • (a + x)
a) Find the zeropoint. Are there points where f is not differentiable?
b) Determine where f increases and decrease, and provide any local and global extremal
and its extreme values.
c) Determine where f is convex and concave, and provide any turning points.

tkhunny · Answer

I suggest you rewrite it a little and worry about the break.  $$f_{1}(x) = x\cdot(a+x)$$ for x > 0 and $$f_{2}(x) = -x\cdot(a+x)$$ for x < 0.  I'll let you consider x = 0.

anonymous · Answer

I found zeropoits at (0,0) and (x=-a,0) , also that f is not differentiable at x=0

anonymous · Answer

the f is increasing for x>0 , but i dont how to proof how f behave for x<0

tkhunny · Answer

Did you determine the 1st Derivative away from x = 0?

anonymous · Answer

i found that f'(x)= |x|+x+a for any x but x=0 if thats right

tkhunny · Answer

Why is the absolute value still in there.  For x < 0, you have a perfectly good definition WITHOUT the aboslute values.

anonymous · Answer

by using a product rule for derivatives

tkhunny · Answer

Do that for ONLY x < 0.  Get rid of the absolute values.