(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A metric on C[0,1] is defined by:

[tex]d(f,g) = ( \int_0^1 \! (f(x) - g_t(x))^2 \, dx )^{1/2}[/tex]

Find t e R such that the distance between the functions [tex]f(x) = e^x - 1[/tex] and [tex]g_t(x) = t * x[/tex] is minimal.

2. Relevant equations

Given above

3. The attempt at a solution

The first thing I did was multiply the inner part of the integral out then evaluate it:

[tex]( \int_0^1 \! (e^x-1) - (tx))^2 dx )^{1/2}[/tex]

[tex]= ( \int_0^1 \! e^{2x} - 2e^x - 2te^xx + 1 + 2tx + t^2x^2 dx )^{1/2}[/tex]

[tex]= (\frac{1}{2}e^2-\frac{1}{2} - (2e - 2) - 2t(e) + 1 + 2t(\frac{1}{2}) + t^2(\frac{1}{3}) )^{1/2}[/tex]

[tex]= ( \frac{1}{2}e^2 + \frac{5}{2} - 2et + t + \frac{1}{3}t^2)^{1/2}[/tex]

But I'm not sure I did that right, because now I don't know where to go from here. Any tips?

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# Homework Help: Minimum distance of functions in a metric space

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