what is true of this function?

Question

f(x)= 9x^3-lnx?
1. domain is all real numbers
2. it is always concave up
3. it has no relative minima
4. it has one relative maximum
5. it has one point of inflection.

i crossed out 3, 4, and 5 because it does have a relative min?
there is no relative max because it always goes up to infinity?
and it seems to have no point of inflection because it is concave up?
is any of this right?
im thinking its number 2? what do you think?

i tried taking the derivative, but what is the derivative of the natural log?
we havnt done early trancendentals yet in my calc class (its in a different chapter than the derivatives)

Answer 1

You differentiated and found it has relative min at x=1/3? but if you differentiate again you find second deriv is
54x - 1/(x^2), and I think this goes from
negative when x<1/cuberoot(54) to positive when it's greater, so that means it has an inflection. Also means it's concave down when
x<1/cuberoot(54), so the answer must be 5, not 2. Unless I made a mistake!

PS Dave is wrong about 1. Domain is all positive real numbers, because lnx is undefined for 0 and neg.

Answer 2

Hi. I know 1 is true, 2 is false (it is more of an "S" shape), 3 is false (x > 0 goes to +infinity, x < 0 goes to -infinity), 4 might be true (infinity?), and I don't know about 5. It's going to act strange around zero.

By the way, if I remember right, the derivative of ln(x) = ln(x)dx (it has been quite a while, tho).

Answer 3

The derivative of the natural log is 1/x. You're almost there.

Answer 4

I think ONE is FALSE as you can not take a log function has a domain of all positive numbers.

Answer 5

number 2 all the way!

Answer 6

you're right...it's 2