Using the Second Derivative Test

[BuBbLeS01]

Homework Statement

Find all relative extrema using the second derivative test for H(x) = x * lnx

Homework Equations

The Attempt at a Solution

H'(x) = (1 * ln x) + (x * 1/x) = lnx + 1
H''(x) = 1/x + 0

Is H''(x) right? Then I am having trouble finding the relative extrema from the second derivative test?

 

[eyehategod]

h''(x)=1/x

 

[BuBbLeS01]

Okay so how do I solve for the relative extrema?

 

[eyehategod]

just try graphing the function and go from there. i don't remember what the second derevative test is, its been a long time. don't you have to set denominator =0 and solve x?if so x=0.

 

[Dick]

Okay so how do I solve for the relative extrema?

Set H'(x)=0 and solve for x to find the critical points. For each critical point x test the sign of H''(x) to see if it's a max or a min.

 

[BuBbLeS01]

H"(x) = 0
I am having trouble finding the relative extrema from the second derivative test?

 

[Dick]

Set H'(x)=0 not H''(x)=0. Critical points are where H'(x)=0.

 

[BuBbLeS01]

How do you solve lnx + 1 = 0 for x?

 

[BuBbLeS01]

would it be...
lnx + 1 = 0
lnx = -1
x = e^-1

 

[BuBbLeS01]

Then I would plug that into H" = 1/x, H" = 1/(e^-1) = e > 0 so its a relative max?

 

[Dick]

That's almost all exactly correct, except the final conclusion. How can you say H''(x)>0 at a critical point means it's a max?? Don't you have like a textbook or something?

 

[BuBbLeS01]

H''(e^-1) > 0 means a relative max...thats what our book says to write? What do you mean? Is that wrong?

 

[HallsofIvy]

Either you should throw away your textbook or you should read it more carefully- that's exactly backwards!

 

[GBK.Xscape]

Yea, for the second derivative tests things are sort of backwards.
If f"(c)<0, then x=c is a relative maximum.
If f"(c)>0, then x=c is a relative minimum.
f f"(c)=0 or undefined, then the second derivative test is inconclusive.