Local minimum question

[madoscar65]

Hi Vners,
Just been stuck on proving this: What values of n for graph of y=x^n to have a local minimum? I know that an even number will have a local minimum but I don't know how to prove it. Any help will be appreciated.

Thanks

Edit: Problem solved

[taiga]

Spent a ridiculous amount of time thinking about this, such a good question!

Glad you've got it, I'll post the solution if anyone wants it, otherwise ceebs

[epinephrine]

Yes I would like to see the solution Taiga 

[taiga]

y = x^n

Assume n is positive.
If n=0 it's just a straight line, for 0<n<1, there are no stationary points, hence there can be no minimum.

dy/dx = n*x^(n-1)

If n is odd, for example 3

dy/dx = n*x^2.
Given n is positive, there is no way here the gradient can be negative due to the squared number.

To have a local minimum, the gradient needs to be negative at some stage then become positive. For any odd number the gradient clearly can not be negative and therefore no local minimum can exist for any odd number.

That's the furthest I can explain it within the scope of maths methods.

EDIT: asked OP and he had a different solution which sounds very reasonable

[madoscar65]

Edit: misread

[taiga]

What do you mean?

I thought you had your own set of solutions that you might have wanted to post (OP in this case means original poster)

[madoscar65]

LOL sorry my bad, misinterpretation