- #1

sergey_le

- 77

- 15

- Homework Statement
- Let function ƒ be Continuous in R, x1 local minimum point of f and x2 local maximum point of f.

Existing f(x1)>f(x2).

Prove that there is another local minimum point f (except x1)

- Relevant Equations
- -

Here's what I tried to do:

f Continuous function at R, x1 local minimum point of f, x2 local maximum point of f.

Existing f(x1)>f(x2).

Let's look at the interval [x1,x2]⊆ℝ .

f is continuous in R and therefore continuous in its partial segment. Therefore f continuous in [x1,x2].

Therefore, there is an environment of δ>0 ,So that for all x that holds x1<x<x1+δ Happening : f(x1)≤f(fx)

Because given that x1 is a local minimum point of f.

And thus to all x that holds x2-δ<x<x2 Happening : f(x)≤f(x2).

Because given that x2 is a local maximum of f.

Don't know how to proceed

f Continuous function at R, x1 local minimum point of f, x2 local maximum point of f.

Existing f(x1)>f(x2).

Let's look at the interval [x1,x2]⊆ℝ .

f is continuous in R and therefore continuous in its partial segment. Therefore f continuous in [x1,x2].

Therefore, there is an environment of δ>0 ,So that for all x that holds x1<x<x1+δ Happening : f(x1)≤f(fx)

Because given that x1 is a local minimum point of f.

And thus to all x that holds x2-δ<x<x2 Happening : f(x)≤f(x2).

Because given that x2 is a local maximum of f.

Don't know how to proceed