Minimizing an Integral

Calculus Level 4

Consider all continuous functions $f:[0,1] \rightarrow \mathbb{R},$ and let $I(f)=\int_{0}^{1} 1+e^{x}f(x)+\big(e^{x}f(x)\big)^{2} \ dx.$ If the minimum value of $I(f)$ across all such functions $f(x)$ is $\frac{m}{n}$ for coprime positive integers $m$ and $n$ , find $m+n.$

Hint: Since we know very little about $f(x)$ , try using algebra to force something inside the integrand that helps you restrict parts of it to known values.

Details and Assumptions:

$I(f)$ takes in a function $f(x)$ and spits out a real number.
You may use the fact that $e \approx 2.718$ .

The answer is 7.

1 solution

Brandon Monsen
Aug 11, 2017

$I(f)=\int_{0}^{1} 1+e^{x}f(x)+[e^{x}f(x)]^{2} \ dx \quad = \quad \int_{0}^{1} \frac{3}{4} + [e^{x}f(x)+\frac{1}{2}]^{2} \ dx \quad \geq \quad \int_{0}^{1} \frac{3}{4} \ dx$

With equality when $f(x)=\frac{-1}{2}e^{-x}$ , which is continuous over $[0,1]$ . Hence, the minimum of $I$ is $\boxed{.75}$

I hosed up my algebra and thus missed out on the solution. You might be interested in researching Pontryagin's Maximum Principal

Pontryagin's maximum principle .

In short this allows us to directly calculate the optimal function. For this example we take the function inside the integral and treat /(f/) as a variable and take the derivative in terms of /(f/), set equal to zero, and solve for /(f/).

This results in

$e^{x}+2e^{2x}f=0$

solving for f we get

$f=\frac{-1}{2}e^{-x}$

And from there we can calculate the minimum value of the integral.

Daniel Branscombe - 3 years, 10 months ago

Very interesting. I'm going to bed right now but I promise I'll look into this tomorrow, and maybe post a harder problem where algebraically finding the minimum is no longer a 3-step solution.

Is this at all related to the Fundamental Theorem of Calculus? I always thought that for that the integral needed bounds as a function of $x$ , but I'm apparently mistaken.

Brandon Monsen - 3 years, 9 months ago

kinda related. This is part of a field of mathematics referred to as Calculus of Variation with the principal I referenced above being just a single example of it.

Here is a good place to get you started

Calculus of variation .

Daniel Branscombe - 3 years, 9 months ago

@Daniel Branscombe – I liked the example with finding the shortest distance between two points, made it very easy to understand what was going on in at least that simple case.

I'm actually taking my upper level classical mechanics course this semester which will go over Lagrangian Mechanics and so I'm sure we will learn this in-depth with principle of least action :)

Brandon Monsen - 3 years, 9 months ago

Minimizing an Integral

The answer is 7.

1 solution

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