Question

How do I apply Weierstrass' theorem to this equation?

Answer 1

\[y=x ^{3}-\ln (x+1)-1\]
How do i know if this function has minimum or a maximum or both by using weierstrass?

Answer 2

do u know wat Weierstrass' theorem exacttly is

Answer 3

no sorry C:

Answer 4

In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function

Answer 5

is this cleared?

Answer 6

ok

Answer 7

But I also remember that it says something about the maximum and the minimum of a function...

Answer 8

The theorem mentioned by @ChiefArnav isn't the only theorem attributed to Weierstrass. The one you want is what's commonly referred to as the extreme value theorem, which says
\[\text{If }f\text{ is a continuous, bounded function in the interval }[a,b],\text{ then}\\
\text{there exist }c,d\text{ in }[a,b]\text{ such that }f(d)\le f(x)\le f(c).\]
In other words, if \(f\) can be restricted to a bounded domain, you can guarantee that the function attains an extremum.

For your function, any such interval would have to be constructed to comply with the function's domain - \(f(x)\) is undefined for \(x\le-1\).

Answer 9

Can you explain the latter part more? because I didn't understand it properly...

Answer 10

The latter part of the theorem, or the last comment about the given function?

Answer 11

The part about the extremum, thanks again

Answer 12

Let's take some generic function \(f(x)\) defined over an interval \([a,b]\):