Reason/Proof Derivative of (ABS(X))/(3X+1) is Undefined at 0

[Barry75]

Hi,

I'm being asked to graph (ABS(X))/(3x+1) using the first and second derivatives. I know, based on the answer key, that it can be split into two piecewise functions of:

1. x/(3x+1), x>= 0
2. -x/(3x+1), x<0

The first derivative, as shown in the answer key, is:
1. 1/(3x+1)² , x>0
2. -1/(3x+1)² , x<0

The answer key states: "Notice that the derivative of the first piece never equals zero and is always positive. The derivative of the second piece never equals zero but is undefined at the asymptote. It is negative everywhere else. However, notice there is a hidden critical point in the derivative. The derivative is undefined at x = 0. That makes this point a critical point. Since the slope is negative on one side and positive on the other, then this point must be a cusp point minimum."

I do not easily notice the hidden critical point in the derivative.... I understand the reason that ABS(X) is not differentiable at 0 and I'm pretty sure that's the reason why in this case but I want to be sure and want to know why.

Before I go into my steps, will any absolute function not be differentiable at the point where the derivative of the part of the function within ABS() is equal to zero? For example, if you had ABS(X²-9), it would not be differentiable at +/- 3. Besides for for completing proof I'm attempting below, is there any other reason I can make the leap that that point at x=0 is not differentiable?

Can someone help complete the proof below? (assuming I'm on the right track)

Instead of of breaking into a piecewise function, I wrote f(x) as (SQRT(X^2))/(3X+1) and tried to prove that as lim x-> 0 of this function that the derivative at x=0 is not the same from the right and the left.

Steps:
1. Wrote out lim x -> 0 (f(x+h) - f(x))/h
2. Multiplied f(x+h) by (3x+1)/(3x+1) and multiplied f(x) by (3(x+h)+1)/(3(x+h)+1)
3. Expanded everything on the numerator of the numerator out

I end up with the following:

((3x(sqrt((x+h)^2)) + sqrt((x+h)^2) - 3x(sqrt(x^2)) - 3h(sqrt(x^2)) - sqrt(x^2))/((3(x+h)+1)(3x+1)))/h

Thanks,
Barry

 

[Dr.Peterson]

The answer key states: "Notice that the derivative of the first piece never equals zero and is always positive. The derivative of the second piece never equals zero but is undefined at the asymptote. It is negative everywhere else. However, notice there is a hidden critical point in the derivative. The derivative is undefined at x = 0. That makes this point a critical point. Since the slope is negative on one side and positive on the other, then this point must be a cusp point minimum."

I do not easily notice the hidden critical point in the derivative.... I understand the reason that ABS(X) is not differentiable at 0 and I'm pretty sure that's the reason why in this case but I want to be sure and want to know why.

I wouldn't really say this critical point is hidden! The absolute value puts a spotlight on it.

Basically, because |x| is not differentiable at x=0, any function containing it is at least suspect at the corresponding point, so you should always check there. To check, you can just find the derivative on the right and on the left and compare them.

Before I go into my steps, will any absolute function not be differentiable at the point where the derivative of the part of the function within ABS() is equal to zero? For example, if you had ABS(X²-9), it would not be differentiable at +/- 3. Besides for for completing proof I'm attempting below, is there any other reason I can make the leap that that point at x=0 is not differentiable?

There are some functions containing an absolute value that turn out to be differentiable at the suspect point because other things cancel it out, so you can't really just assume it is not. But it is not a "leap" to say that it might not be.

Can someone help complete the proof below? (assuming I'm on the right track)

Instead of of breaking into a piecewise function, I wrote f(x) as (SQRT(X^2))/(3X+1) and tried to prove that as lim x-> 0 of this function that the derivative at x=0 is not the same from the right and the left.

Steps:
1. Wrote out lim x -> 0 (f(x+h) - f(x))/h
2. Multiplied f(x+h) by (3x+1)/(3x+1) and multiplied f(x) by (3(x+h)+1)/(3(x+h)+1)
3. Expanded everything on the numerator of the numerator out

I end up with the following:

((3x(sqrt((x+h)^2)) + sqrt((x+h)^2) - 3x(sqrt(x^2)) - 3h(sqrt(x^2)) - sqrt(x^2))/((3(x+h)+1)(3x+1)))/h

Thanks,
Barry

This would be a lot of work to complete. I think the piecewise approach is far more sensible. I would never go to the definition of the derivative unless it was required!

 

[Barry75]

That makes sense. Thanks again Dr.Peterson.