The number of points in the interval [2, 4], | vedclass

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(D) Let $g(x) = x^2 - x - 0.5$. The function $f(x) = [g(x)]$ is discontinuous at points where $g(x)$ takes an integer value.We need to find the number

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$10$

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(D) Let $g(x) = x^2 - x - 0.5$. The function $f(x) = [g(x)]$ is discontinuous at points where $g(x)$ takes an integer value.We need to find the number of points $x \in [2, 4]$ such that $g(x) = k$, where $k$ is an integer.First, evaluate the range of $g(x)$ on the interval $[2, 4]$.$g(2) = 2^2 - 2 - 0.5 = 1.5$.$g(4) = 4^2 - 4 - 0.5 = 11.5$.Since $g'(x) = 2x - 1$, for $x \in [2, 4]$, $g'(x) > 0$, so $g(x)$ is strictly increasing.As $x$ varies from $2$ to $4$, $g(x)$ takes all values in the interval $[1.5, 11.5]$.The integer values $k$ that $g(x)$ takes in this interval are $k \in \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$.For each such integer $k$, there exists exactly one $x \in [2, 4]$ such that $g(x) = k$ because $g(x)$ is strictly increasing.The number of such integers is $11 - 2 + 1 = 10$.Thus, there are $10$ points of discontinuity in the interval $[2, 4]$.

The number of points in the interval $[2, 4]$, at which the function $f(x) = [x^2 - x - 1/2]$, where $[·]$ denotes the greatest integer function, is discontinuous, is ————

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