Let the function \(f(x) = (x^2 - 1)|x^2 - ax + 2| + \cos|x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to:
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For absolute value functions, points of non-differentiability occur where the expression inside the absolute value equals zero.
Step 1: Identifying non-differentiable points.
\(\cos|x|\) is always differentiable. Therefore, we only need to check where \(|x^2 - ax + 2|\) is not differentiable.
Equating the inside expression to zero:
\[
x^2 - ax + 2 = 0
\]
Since one root is given as \( \alpha = 2 \), substituting this value:
\[
4 - 2a + 2 = 0 \implies a = 3
\]
With \(a = 3\), the other root becomes \( \beta = 1\).
Step 2: Finding the distance from the line.
The point \((\alpha, \beta) = (2, 1)\).
Using the point-to-line distance formula:
\[
d = \frac{|12(2) + 5(1) + 10|}{\sqrt{12^2 + 5^2}} = \frac{|24 + 5 + 10|}{\sqrt{144 + 25}} = \frac{39}{\sqrt{169}} = \frac{39}{13} = 3
\]
