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Let y = f(x) be defined in [a, b], then ...

Let y = f(x) be defined in [a, b], then
(i) Test of continuity at `x = c, a lt c lt b`
(ii) Test of continuity at x = a
(iii) Test of continuity at x = b
Case I Test of continuity at `x = c, a lt c lt b`
If y = f(x) be defined at x = c and its value f(c) be equal to limit of f(x) as `x rarr c` i.e. f(c) = `lim_(x to c) f(x)`
or `lim_(x to c^(-))f(x) = f(c) = lim_(x to c^(+)) f(x)`
or LHL = f(c) = RHL
then, y = f(x) is continuous at x = c.
Case II Test of continuity at x = a
If RHL = f(a)
Then, f(x) is said to be continuous at the end point x = a
Case III Test of continuity at x = b, if LHL = f(b)
Then, f(x) is continuous at right end x = b.
Number of points of discontinuity of `[2x^(3) - 5]` in [1, 2) is (where [.] denotes the greatest integral function.)

A

14

B

13

C

10

D

None of these

Text Solution

AI Generated Solution

The correct Answer is:
To determine the number of points of discontinuity of the function \( y = \lfloor 2x^3 - 5 \rfloor \) in the interval \([1, 2)\), we will follow these steps: ### Step 1: Analyze the function The function \( y = 2x^3 - 5 \) is a continuous function on the interval \([1, 2)\). However, since we are interested in the greatest integer function (denoted by \( \lfloor \cdot \rfloor \)), we need to find the points where \( 2x^3 - 5 \) takes integer values, as the greatest integer function is discontinuous at integer values. ### Step 2: Determine the range of \( 2x^3 - 5 \) over \([1, 2)\) We will evaluate the function at the endpoints of the interval: - At \( x = 1 \): \[ 2(1)^3 - 5 = 2 - 5 = -3 \] - At \( x = 2 \): \[ 2(2)^3 - 5 = 2(8) - 5 = 16 - 5 = 11 \] Thus, the function \( 2x^3 - 5 \) ranges from \(-3\) to \(11\) as \( x \) varies from \( 1 \) to \( 2 \). ### Step 3: Identify the integer values in the range The integers in the range from \(-3\) to \(11\) are: \[ -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \] ### Step 4: Count the integers Now, we will count the integers: - Starting from \(-3\) to \(10\), we have: - \(-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\) Counting these gives us a total of \(14\) integers. ### Conclusion The number of points of discontinuity of \( y = \lfloor 2x^3 - 5 \rfloor \) in the interval \([1, 2)\) is \(14\).
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