The number of points in the interval [2. 4], at which the function , where [] denotes the greatest integer function, is discontinuous, is ________. [2026]

Question

The number of points in the interval [2. 4], at which the function $f (x) = [x^{2} - x - \frac{1}{2}]$ , where [ $\cdot$ ] denotes the greatest integer function, is discontinuous, is ________. [2026]

Smart Achievers · Accepted Answer

(10)

We have, $f (x) = [x^{2} - x - \frac{1}{2}]$

At x = 2, $f (x) = [4 - 2 - \frac{1}{2}] = [1.5]$

At x = 4, $f (x) = [16 - 4 - \frac{1}{2}] = [11.5]$

Now, let $g (x) = x^{2} - x - \frac{1}{2}$

$g' (x) = 2 x - 1$

$∴$ g(x) is strictly increasing in [2, 4]

$∴$ 1.5 < k < 11.5

k $\in$ [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

$∴$ No. of discontinuity points = 10.

Solution

Q.
The number of points in the interval [2. 4], at which the function $f (x) = [x^{2} - x - \frac{1}{2}]$ , where [ $\cdot$ ] denotes the greatest integer function, is discontinuous, is ________. [2026]

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Solution

Q. The number of points in the interval [2. 4], at which the function f(x)=[x2–x–12], where [·] denotes the greatest integer function, is discontinuous, is ________. [2026]

Ans. (10) We have, f(x)=[x2–x–12] At x = 2, f(x)=[4–2–12]=[1.5] At x = 4, f(x)=[16–4–12]=[11.5] Now, let g(x)=x2–x–12 g'(x)=2x–1 ∴ g(x) is strictly increasing in [2, 4] ∴ 1.5 < k < 11.5 k∈[2, 3, 4, 5, 6, 7, 8, 9, 10, 11] ∴ No. of discontinuity points = 10.

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Q.
The number of points in the interval [2. 4], at which the function $f (x) = [x^{2} - x - \frac{1}{2}]$ , where [ $\cdot$ ] denotes the greatest integer function, is discontinuous, is ________. [2026]