Inequality Problem: Prove f^2 ≤ 1/4 (f')^2

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In summary, the problem is to prove that the integral of f^2 on [0,1] is less than or equal to 1/4 times the integral of f' squared on [0,1], given that f is a continuously differentiable function on [0,1] and f(0)=f(1)=0. One possible approach is to use the fact that f(0)=0 gives a formula for f(x) involving an integral from 0 to x, and f(1)=0 gives a formula for f(x) involving an integral from x to 1. This is similar to Wirtinger's Inequality, which also involves integrals of functions and their derivatives. Further discussion can be found
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Homework Statement


Let [tex]f\in C^{1}[/tex] on [0,1], and [tex]f(0)=f(1)=0[/tex], prove that
[tex]\int_{0}^{1}(f(x))^{2}dx \leq \frac{1}{4} \int_{0}^{1} (f'(x))^{2}dx[/tex]


Homework Equations





The Attempt at a Solution


what is the trick to produce a 1/4 there? and how to make use of f(0)=f(1)=0? well I know that f(0)=0 gives a formula like [tex]f(x)=\int_{0}^{x} f'(t)dt[/tex] and f(1)=0 gives [tex]f(x)= -\int_{x}^{1} f'(t)dt[/tex]. But seems that I cannot go further.
Any small hint would be great, thanks!
 
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  • #2
anyone?...
 
  • #3
Isn't this Wirtinger's Inequality?
 
  • #5
yea, I'm reading it. cool. Thanks guys!
 

Related to Inequality Problem: Prove f^2 ≤ 1/4 (f')^2

1. What is the inequality problem f^2 ≤ 1/4 (f')^2?

The inequality problem f^2 ≤ 1/4 (f')^2 is a mathematical inequality that relates to the behavior of a function f and its derivative f'. It states that the square of the function f is always less than or equal to one-fourth of the square of its derivative f'.

2. What does this inequality prove?

This inequality proves that the function f and its derivative f' have a certain relationship in terms of their magnitudes. More specifically, it shows that the magnitude of f is always smaller than the magnitude of f' multiplied by a constant factor of 1/2.

3. What is the significance of this inequality in science?

This inequality has significance in many scientific fields, such as physics, engineering, and economics. It is commonly used to analyze the stability and behavior of systems, as well as to make predictions about future trends. In economics, it can be applied to study income inequality and wealth distribution.

4. How is this inequality proven?

This inequality is proven using mathematical techniques, such as algebra and calculus. The proof involves manipulating the equation until it takes the form of f^2 ≤ 1/4 (f')^2. This can be done by using the properties of derivatives and basic algebraic operations.

5. What are some real-life examples of this inequality?

One example of this inequality can be seen in the study of population growth. The function f represents the population size at a given time, while the derivative f' represents the rate of change of the population. This inequality shows that the population size will always be smaller than one-fourth of the rate of change of the population. Other examples include the relationship between distance and velocity in physics, and the relationship between income and savings in economics.

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