Question

M = {x / x ∈ IN , -2 < x < 5}


alguien sabe resolverlo?

Answer 1

Respuesta:

Problem. Let f : R → R be a differentiable function such that |f

0

(x)| ≤ π/6 for all

x ∈ R. If f(0) = ln(5) what are the possible values of f(e)?

Solution. It must be that f(e) ∈ [−πe

6 + ln(5),

πe

6 + ln(5)]. If f(e) does not belong to

this interval, then

 

 

f(e) − f(0)

e − 0

 

 > π/6

which means that there is x ∈ [0, e] such that |f

0

(x)| > π/6 by the Mean Value Theorem.

Conversely, by picking m ∈ [−π/6, π/6] the function f(x) = ln(5) + mx can be made to

achieve at f(e) any value in [−πe

6 + ln(5),

πe

6 + ln(5)

Explicación paso a paso: