Let $f(x)=4x^2-4ax+a^2-2a+2$ be a quadratic polynomial in $x$, $a \in R$. If at least one root of $f(x)=0$ lies in $[0,2]$, then the value of $a$ belongs too:

To find the values of a for which at least one root of f(x) = 4x^2 - 4ax + a^2 - 2a + 2 = 0 lies in the interval [0, 2], we use the intermediate value property by ensuring f(0) * f(2) <= 0. Evaluating f(0) gives a^2 - 2a + 2, which has a discriminant of 4 - 8 = -4, meaning it is always positive. Evaluating f(2) gives 16 - 8a + a^2 - 2a + 2, which simplifies to a^2 - 10a + 18. The condition f(0) * f(2) <= 0 simplifies to a^2 - 10a + 18 <= 0. Solving the quadratic inequality a^2 - 10a + 18 = 0 gives a = (10 +/- sqrt(100 - 72))/2 = 5 +/- sqrt(7). Therefore, the value of a belongs to the interval [1, 5 - sqrt(7)] union [5 + sqrt(7), infinity), which matches the choice [1, 5 - sqrt(7)] combined with the fact that for the root to be in [0,2], a must fall into the required range.

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