50 puntos?

Tarea : Sea la función<img src="https://tex.z-dn.net/?f=F%3A%5B0%2C%2B%5Cinfty%29%5Clongrightarrow%5Cmathbb%7BR%7D%2C%20F%28x%29%3D%5Cdfrac%7B1%7D%7Bx%2B3%7D-%5Cdfrac%7B1%7D%7Bx%2B4%7D" /> a) Determina la función f : (0, + ∞)→R, tal que F sea una primitiva para la f.

B) Demuestre que F es decreciente en [0, + ∞) c) Demuestre que<img src="https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B20%7D%5Cleq%5Cint%5Climits%5E1_0%20%7BF%28x%29%7D%20%5C%2C%20dx%20%5Cleq%20%5Cfrac%7B1%7D%7B2%7D" /> Respuesta : a) La función f(x) tal que F(x) es la primitiva es : 1 / (x + 4)² - 1 / (x + 3)² b) Verdadero.

C) Verdadero.

Explicación paso a paso : Recordemos que F(x) es primitiva de f(x) si dF(x) / dx = f(x), entonces la primitiva de f(x) será F'(x).

<img src="https://tex.z-dn.net/?f=F%27%28x%29%3D%28%5Cdfrac%7B1%7D%7Bx%2B3%7D%29%27-%28%5Cdfrac%7B1%7D%7Bx%2B4%7D%29%27%3D%5Cdfrac%7B%281%29%27%28x%2B3%29-%281%29%28x%2B3%29%27%7D%7B%28x%2B3%29%5E%7B2%7D%7D-%5Cdfrac%7B%281%29%27%28x%2B4%29-%281%29%28x%2B4%29%27%7D%7B%28x%2B4%29%5E%7B2%7D%7D%5C%5C%5C%5C%3D%5Cdfrac%7B0-1%7D%7B%28x%2B3%29%5E%7B2%7D%7D-%5Cdfrac%7B0-1%7D%7B%28x%2B4%29%5E%7B2%7D%7D%3D-%5Cdfrac%7B1%7D%7B%28x%2B3%29%5E%7B2%7D%7D%2B%5Cdfrac%7B1%7D%7B%28x%2B4%29%5E%7B2%7D%7D" /> La función es decreciente si su derivada es < 0 : <img src="https://tex.z-dn.net/?f=F%27%28x%29%3D-%5Cdfrac%7B1%7D%7B%28x%2B3%29%5E%7B2%7D%7D%2B%5Cdfrac%7B1%7D%7B%28x%2B4%29%5E%7B2%7D%7D%3D-%5Cdfrac%7B1%7D%7Bx%5E%7B2%7D%2B3x%2B9%7D%2B%5Cdfrac%7B1%7D%7Bx%5E%7B2%7D%2B8x%2B16%7D%5C%5C%5C%5C%3D%5Cdfrac%7B1%7D%7Bx%5E%7B2%7D%2B8x%2B16%7D-%5Cdfrac%7B1%7D%7Bx%5E%7B2%7D%2B3x%2B9%7D%3D%5Cdfrac%7Bx%5E%7B2%7D%2B3x%2B9-x%5E%7B2%7D-8x-16%7D%7B%28x%2B3%29%5E%7B2%7D%2A%28x%2B4%29%5E%7B2%7D%7D%5C%5C%5C%5C%3D%5Cdfrac%7B-5x-7%7D%7B%28x%2B3%29%5E%7B2%7D%2A%28x%2B4%29%5E%7B2%7D%7D%3D-%5Cdfrac%7B5x%2B7%7D%7B%28x%2B3%29%5E%7B2%7D%2A%28x%2B4%29%5E%7B2%7D%7D" />F'(x) es < 0 para todo su dominio, incluyendo [0, + ∞) Para demostrar la desigualdad conviene calcular primero la integral de en medio : <img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B1%7D%20F%28x%29%5C%2C%20dx%3D%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B1%7D%20%5Cdfrac%7B1%7D%7Bx%2B3%7D%5C%2C%20dx-%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B1%7D%20%5Cdfrac%7B1%7D%7Bx%2B4%7D%5C%2C%20dx" />Estas integrales se pueden resolver con sustituciones : u = x + 3 , du = dxv = x + 4 , dv = dx<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B1%7D%20%5Cdfrac%7B1%7D%7Bx%2B3%7D%5C%2C%20dx-%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B1%7D%20%5Cdfrac%7B1%7D%7Bx%2B4%7D%5C%2C%20dx%3D%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B1%7D%20%5Cfrac%7Bdu%7D%7Bu%7D%20%5C%2C%20dx-%5Cdisplaystyle%5Cint_%7B0%7D%5E%7B1%7D%20%5Cfrac%7Bdv%7D%7Bv%7D%20%5C%2C%20dx%3D%20%7B%5Cleft%20%5Cln%28u%29%20%5Cright%7C%7D_%7B0%7D%5E%7B1%7D-%7B%20%5Cleft%20%5Cln%28v%29%20%5Cright%7C%7D_%7B0%7D%5E%7B1%7D%3D%7B%20%5Cleft%20%5Cln%28x%2B3%29%20%5Cright%7C%7D_%7B0%7D%5E%7B1%7D-%7B%20%5Cleft%20%5Cln%28x%2B4%29%20%5Cright%7C%7D_%7B0%7D%5E%7B1%7D%3D%5Cln%281%2B3%29-%5Cln%281%2B4%29-%28%5Cln%280%2B3%29-%5Cln%280%2B4%29%29%3D0%2C06" />Luego : 1 / 20 ≤ 0, 06 ≤ 1 / 20, 05 ≤ 0, 06 ≤ 0, 5.