Two true/false questions I don't understand

[IntegrateMe]

1. If a is a positive, then the function h(x) = (ln(ax^2)+x)/x is an antiderivative of j(x) = (2-ln(ax^2))/x^2

So, I used Wolfram and took the integral of j(x) with different values for a and always got ln(ax^2)/x, so I put false. However, the answer is true, and I can't figure out why!

2. If x = a is a critical point of a function m(x), then m'(a) = 0.

For this one I put true, and the answer is false. Is it because m'(a) can also be undefined?

Thank you!

 

[Dick]

The difference between (ln(ax^2)+x)/x and ln(ax^2)/x is a constant. What constant? So they are both antiderivatives of (2-ln(ax^2))/x^2. And sure, m'(a) might be undefined at a critical point.

 

[IntegrateMe]

How about this:

If f"(a) = 0, then f has an inflection point at x = a.

The answer is false, but I thought inflection points were where the second derivative is equal to 0? Any clarification on this one?

 

[Dick]

How about this:

If f"(a) = 0, then f has an inflection point at x = a.

The answer is false, but I thought inflection points were where the second derivative is equal to 0? Any clarification on this one?

Same deal. An inflection point is where the concavity changes from concave up to concave down or vice versa. Define f(x)=x^2 for x>=0 and -x^2 for x<0. The derivative exists and is continuous and x=0 is an inflection point, but the second derivative doesn't exist there.

 

[IntegrateMe]

But this problem is explicitly saying that the second derivative is equal to 0 at a. So shouldn't a on the original graph be an inflection point?

 

[Dick]

But this problem is explicitly saying that the second derivative is equal to 0 at a. So shouldn't a on the original graph be an inflection point?

Yeah, I was going backwards. Think about f(x)=x^4. Is x=0 an inflection point?

 

[IntegrateMe]

No, it's not! I can see what you mean graphically, but can you explain it more clearly? I'm sorry I didn't follow the first time.

 

[Dick]

No, it's not! I can see what you mean graphically, but can you explain it more clearly? I'm sorry I didn't follow the first time.

I mean that if f(x)=x^4 then f''(x)=0. But f(x) is concave up everywhere.

 

[IntegrateMe]

So if we're given a function and asked to find inflection points, how can we verify that it's an inflection point besides checking that the second derivative is equal to 0? Do we have to check the first derivative and see if it changes from +/- or -/+ at that point as well?

 

[Dick]

So if we're given a function and asked to find inflection points, how can we verify that it's an inflection point besides checking that the second derivative is equal to 0? Do we have to check the first derivative and see if it changes from +/- or -/+ at that point as well?

No, I think you have to check if the second derivative changes sign.