# 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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