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

In summary, the conversation is about finding the maximum value of a function and calculating the limit of a sequence. The first part involves solving an equation using derivatives, while the second part discusses the concept of critical points and endpoints in finding the maximum value.
  • #1
help
14
0
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
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  • #2
help said:
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)?
 
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Likes Delta2 and member 587159
  • #3
Help, you could use the product rule, then set it equal to zero.
 
  • #4
help said:
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.
 

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

1. What is the purpose of differentiating a function?

Differentiating a function allows us to find the rate of change of the function at a given point. It helps us understand how the function is changing and can be useful in solving optimization problems.

2. How do you differentiate a function using the power rule?

The power rule states that to differentiate a function of the form f(x) = x^n, we bring down the exponent n and subtract 1 from the exponent. In the case of f(x) = x(1-x)^n, we would use the product rule to differentiate the function.

3. Can you explain the product rule for differentiating a function?

The product rule states that to differentiate a product of two functions, we take the derivative of the first function and multiply it by the second function, then add the derivative of the second function multiplied by the first function. In the case of f(x) = x(1-x)^n, we would differentiate x and (1-x)^n separately, then use the product rule to combine them.

4. How does the exponent n affect the differentiation of the function?

The exponent n affects the differentiation of the function by changing the power rule to the product rule. The higher the value of n, the more terms will be involved in the product rule, making the differentiation process more complex.

5. What is the significance of the function f(x) = x(1-x)^n in scientific research?

This function is commonly used in mathematical models to represent processes with limited resources or competition. It can be used to model population growth, chemical reactions, and other natural phenomena. By differentiating this function, we can analyze the rate of change of these processes and make predictions about their behavior.

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