Maths question - can this equation be solved algebraically?

[ultrafilter]

@Hyperelastic

x[sup]x[/sup] is an interesting function. It has a minimum on x>0. What is the value of the minimum and where does it occur?

At x = 1/e, where x^x = 1/e[sup]1/e[/sup].

[OldGuy]

@Hyperelastic

x[sup]x[/sup] is an interesting function. It has a minimum on x>0. What is the value of the minimum and where does it occur?

It’s minimum occurs at x = 1/e =(apprx) 0.367879. The value at the minimum is approximately 0.692201.

It’s simplest to find by noting that the minimum of f(x) and log(f(x)) occur at the same value of x provided f(x) > 0 so for f(x) = x^x, log(f(x)) = xlog x. The derivative of xlog(x) is log(x) + x/x= log(x)+1 . so we want log(x) = -1 which is 1/e.

[Hyperelastic]

@OldGuy

It’s minimum occurs at x = 1/e =(apprx) 0.367879. The value at the minimum is approximately 0.692201.

It’s simplest to find by noting that the minimum of f(x) and log(f(x)) occur at the same value of x provided f(x) > 0 so for f(x) = x^x, log(f(x)) = xlog x. The derivative of xlog(x) is log(x) + x/x= log(x)+1 . so we want log(x) = -1 which is 1/e.

Excellent. Another way to do it is to use the identity x^x=exp(x ln x), then use the elementary rules of differentiation.

[Orbifold]

@Jinx

The academic convention is to blind us with algebra and logic, but guess what? They hide from us the fact that some problems are solved by iterations. One well-known method for this is Newton’s method. So, iterations have been a tool of the trade for a long time, but our “high-and-mighty” academics fail to teach us to think outside the box and there’s more than one way to skin a cat. We need to teach people such thinking because it is just as valuable when tackling problems.

• Jinx

I’m a “high-and-mighty” academic, and my lesson plan includes teaching my students Newton’s Method this coming Tuesday. Come on by, maybe you’ll learn something.

[Yeticus_Rex]

@Orbifold

I’m a “high-and-mighty” academic, and my lesson plan includes teaching my students Newton’s Method this coming Tuesday. Come on by, maybe you’ll learn something.

Can this old rusty tool show up to your class? What time is the class, and can I just buy a parking pass and just sit in the back of the class? I’m 50 miles away from UC Riverside. I’m hoping it’s an evening class.

[Orbifold]

@Yeticus Rex

Can this old rusty tool show up to your class?

blinks

I must confess: despite my flippant response to Jinx’s ill-informed tirade, I don’t really have the authority to issue invitations. The class is for the benefit of the paying students.

That said, if someone sits quietly in the back, doesn’t annoy or harass the students, and doesn’t take a seat from a registered student, it’s no skin off my nose. If you’re serious, send me an e-mail.

(And please understand, I’m not trying to be hostile to the idea. It’s just that my position obligates me to be very, very careful that my responbilities to my students are adequately carried out. Working with students is a bit of a minefield, even when the students are technically adults.)

What time is the class, and can I just buy a parking pass and just sit in the back of the class? I’m 50 miles away from UC Riverside. I’m hoping it’s an evening class.

Oooh, there I have to disappoint you. I teach at 8:10 in the morning.