Finding the Maximum Value of f(x)=x(1-x)^n in [0,1]: A Calculus Problem

[help]

Homework Statement

help with my work

Relevant Equations

derivative and integral

Hello, good afternoon guys. I need a little help from you: D

1)n a positive integer.
f(x)=x(1-x)^n
Solve the equation f'(x)=0 in 0<x<1

this question I asked and found x = 1 / (x + 1) ok

2)An be the maximum value of f(x)=x(1-x)^n in [0,1]
Calculate lim (n+1)an
n tending to infinity

the maximum value would not be making the derivative equal to zero? would it be x = 1? I did not understand

 

[nrqed]

Homework Statement:: help with my work
Relevant Equations:: derivative and integral

Hello, good afternoon guys. I need a little help from you: D

1)n a positive integer.
f(x)=x(1-x)^n
Solve the equation f'(x)=0 in 0<x<1

this question I asked and found x = 1 / (x + 1) ok

2)An be the maximum value of f(x)=x(1-x)^n in [0,1]
Calculate lim (n+1)an
n tending to infinity

the maximum value would not be making the derivative equal to zero? would it be x = 1? I did not understand View attachment 264068

You did not do the first part correctly.

What is the derivative f'(x)?

 

[osilmag]

Help, you could use the product rule, then set it equal to zero.

 

[vela]

Homework Statement:: help with my work
Relevant Equations:: derivative and integral

the maximum value would not be making the derivative equal to zero? would it be x = 1? I did not understand

The maximum value might be attained at a critical point or at the endpoints of the closed interval.

Try plotting the function for a few values of $n$ to get an idea of what you should get.