Question

f(x)=xln(x)
1. Find all critical numbers.
2. Find where the function is increasing and decreasing.
3. Find critical points and identify each as a relative maximum, relative minimum, or neither.
4. Find second order critical numbers and tell where the graph is concave up and where it is concave down.
5. Sketch the graph.

Answer 1

Well, critical points as well as increasing and decreasing require us to get the first derivative. This will be a product rule. Are we okay with derivatives, or should we go over taking the derivative for this problem?

Answer 2

The first derivative is 1+lnx. How to find the critical numbers?

Answer 3

We need to set the derivative equal to 0. So set 1 + lnx = 0 and solve for x.

Answer 4

Okay! Then we get lnx=-1. What is x?

Answer 5

@mathsabc Now turn it into exponential form..\[\bf \ln(x) = a \implies e^a = x\]

Answer 6

Right, genius showed ya, lol.

Answer 7

Oh.. my mistake. I forgot to do this. :D

Answer 8

Mhm. So now you have x = e^(-1). So this would be your critical point. Now to check increasing and decreasing, we need to choose a point on each side of our critical point and plug those in for x in the DERIVATIVE, not the original. So choose two points, one on each side of e^-1 and then plug them in for xin your derivative.

Answer 9

@Psymon You might as well check f"(e^-1) = 1/(e^-1) which would tell you whether the function is concave up/down depending on if f''(x) > 0 or f''(x) < 0..

Answer 10

Might as well. I guess I was just going in order, lol. 1st derivative stuff then 2nd derivative stuff xD

Answer 11

Clearly f''(1/e) = e > 0 hence function is concave up...Therefore it's decreasing to the left of x = 1/e and increasing to the right of x = 1/e..

Answer 12

Sorry.....

Answer 13

That really helped. I know the steps that follow finding value of x. I was stuck at only finding x. My silly mistake.

Answer 14

Eh, I'll stop doing these problems then, I don't want to give people bad info >.<