extreme points of the function

Here is the graph?

If you want to differentiate, you can try logarithmic differentiation.

Do you know how to apply that?.
 

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I tried, haha.
I know how to do it once, for just x^x

but I'm getting confused with this one because the exponent is x^x again

If you take the ln of both sides
lny = lnx^(x^x)
Then bring the exponent down
lny = (x^x)lnx
Then take the derivative?
1/y y' = ?
How would you do that? Do you have to do logarithmic differentiation a second time?
 
I get:

y=x(xx)\displaystyle y=x^{(x^{x})}

ln(y)=ln(x(xx))\displaystyle ln(y)=ln(x^{(x^{x})})

ln(y)=xxln(x)\displaystyle ln(y)=x^{x}ln(x)

Differentiate:

yy=xx1(x(ln(x))2+xln(x))\displaystyle \frac{y'}{y}=x^{x-1}\left(x(ln(x))^{2}+xln(x)\right)

But, y=x(xx)\displaystyle y=x^{(x^{x})}

Therefore, solving for y', we get:

y=xxx+x1(x(ln(x))2+xln(x)+1)\displaystyle \boxed{y'=x^{x^{x}+x-1}\left(x(ln(x))^{2}+xln(x)+1\right)}

This will be difficult to set to 0 and solve for x. Maybe try Newton's method. Or better yet, technology.

By looking at the graph, it doesn't appear to have many extrema. I tried Newton with various values and it didn't want to converge well for any of them.
 
Since x > 0 (given), the domain of the function is (0,?). Looking at the above graph of the function, it seems that no extrema exist, -- extrema being the minimum and/or maximum of a function on an interval.

f ' (x) = x^(x^2+x-1)[1+xln(x)+x[ln(x)]^2}. Set equal to zero. Since x raise to any power is always greater than zero, we only have 1+xln(x)+x[ln(x)]^2 =0. 1+xln(x)+x[ln(x)]^2 is always >0, hence, no extrema exist.
 
f ' (x) = x^(x^2+x-1)[1+xln(x)+x[ln(x)]^2}. Set equal to zero. Since x raise to any power is always greater than zero, we only have 1+xln(x)+x[ln(x)]^2 =0. 1+xln(x)+x[ln(x)]^2 is always >0, hence, no extrema exist.

Good observation. :D

I was certainly maker it tougher than need be talking about Newton and all that...duh.
 
DR._Glockman said:
we only have 1+xln(x)+x[ln(x)]^2 =0. 1+xln(x)+x[ln(x)]^2 is always >0, hence, no extrema exist.

I see that it's always greater than zero, but why does that mean no extrema exist?
 
Ohhhhh
Okay, just kidding I understand now, I wasn't thinking.
if its always greater than zero, it can never equal zero, so no critical points, no extrema...

But could you show me the steps on how you get the derivative?
DR._Glockman said:
f ' (x) = x^(x^2+x-1)[1+xln(x)+x[ln(x)]^2}.
 
But could you show me the steps on how you get the derivative?
DR._Glockman said:
f ' (x) = x^(x^2+x-1)[1+xln(x)+x[ln(x)]^2}.



I thought I showed that in my first post. What part do you not understand?.

To find the derivative of xx\displaystyle x^{x} you can proceed as follows.

y=xx\displaystyle y=x^{x}

ln(y)=ln(xx)=xln(x)\displaystyle ln(y)=ln(x^{x})=xln(x)

yy=ln(x)+x1x=ln(x)+1\displaystyle \frac{y'}{y}=ln(x)+x\cdot\frac{1}{x}=ln(x)+1

y=y(ln(x)+1)=xx(ln(x)+1)\displaystyle y'=y(ln(x)+1)=x^{x}(ln(x)+1)

Now, use that with the product rule to find the derivative of xxln(x)\displaystyle x^{x}ln(x)

xx1x+ln(x)(xx(ln(x)+1))=xx1+ln(x)(xx(ln(x)+1))=xx1(x(ln(x))2+xln(x)+1)\displaystyle x^{x}\cdot\frac{1}{x}+ln(x)(x^{x}(ln(x)+1))=x^{x-1}+ln(x)(x^{x}(ln(x)+1))=x^{x-1}\left(x(ln(x))^{2}+xln(x)+1\right)

Now, when we multiply that by xxx\displaystyle x^{x^{x}} from the original problem above, we get the result we showed.
 
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