Minimizing an Integral

Calculus Level 5

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 \(e \approx 2.718\).

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