True or false on derivatives.

Question

We were given 20 statements to tell if they were true or false and explain why. I have questions on four of them because I am really unsure on my logic here.

1)The derivative of f(x)=1/x² is f'(x)=1/2x

I said this was false because you cant take a derivative using the power rule when it is in the denominator correct?

2) The derivative of π/x² is -π/x. I said this was wrong because the derivative would have a 2 in it somewhere and it wouldn't be negative right?

3) f(x)+g(x) is concave up for all of x. I said this was true, but I don't know why, it just didn't seem false. Could you explain why it is or isn't?

4) f(x)-g(x) cannot be concave up for all of x. I said this was also because the word cannot, but I have no reason for it. Could you please explain this one as well.

I'm not sure on these, but could you please help clear up my logic and explain them. I would be grateful.

Answer 1

1) and 2)

1/x² = x^-2 and π/x² = πx^-2

3) and 4)

The concavity of f(x) and g(x) is not known.

Therefor the sum or difference of the two is not known.