Question

The number of points in the interval \( [2, 4] \), at which the function \( f(x) = \left\lfloor x^2 - x - \frac{1}{2} \right\rfloor \), where \( \left\lfloor \cdot \right\rfloor \) denotes the greatest integer function, is discontinuous, is _______.

Answer 1

Correct Answer: 3

Step 1: Understanding the Concept:
A function of the form \(f(x) = [g(x)]\), where \([\cdot]\) is the greatest integer function, is generally discontinuous at all points where \(g(x)\) takes an integer value.
We must analyze the behavior of the inner function \(g(x)\) over the given interval.
Step 2: Key Formula or Approach:
Let \(g(x) = x^2 - x - 0.5\).
Find the derivative \(g'(x)\) to check for monotonicity.
Determine the range of \(g(x)\) on the interval \([2, 4]\) by evaluating the endpoints.
Count the number of distinct integers within this range.
Step 3: Detailed Explanation:
The inner function is \(g(x) = x^2 - x - 0.5\).
Differentiating it, we get \(g'(x) = 2x - 1\).
For \(x \in [2, 4]\), \(g'(x) \ge 2(2) - 1 = 3>0\).
Since \(g'(x)>0\), the function \(g(x)\) is strictly increasing over the interval \([2, 4]\).
Evaluate \(g(x)\) at the endpoints of the interval:
\[ g(2) = 2^2 - 2 - 0.5 = 4 - 2.5 = 1.5 \] \[ g(4) = 4^2 - 4 - 0.5 = 16 - 4.5 = 11.5 \] Since \(g(x)\) is continuous and strictly increasing from \(1.5\) to \(11.5\), it will cross every integer value between \(1.5\) and \(11.5\) exactly once.
The integer values within this range are: \(2, 3, 4, 5, 6, 7, 8, 9, 10, 11\).
Each time \(g(x)\) hits one of these integers, the greatest integer function \([g(x)]\) will have a jump discontinuity.
There are exactly 10 such integer values.
Step 4: Final Answer:
The number of points of discontinuity is \(10\).