Question

Let f: R → R be a function defined by:

f(x) = `{{:(max_(t ≤ x){t^3 - 3t},;, x ≤ 2),(x^2 + 2x - 6,;, 2 < x ≤ 3),([x - 3] + 9,;, 3 < x ≤ 5),(2x + 1,;, x > 5):}`

where [t] is the greatest integer less than or equal to t. Let m be the number of points where f is not differentiable and I = `int_-2^2f(x)dx`. Then the ordered pair (m, I) is equal to ______.

पर्याय

• `(3, 27/4)`

• `(3, 23/4)`

• `(4, 27/4)`

• `(4, 23/4)`

Answer 1

Let f: R → R be a function defined by:

f(x) = `{{:(max_(t ≤ x){t^3 - 3t},;, x ≤ 2),(x^2 + 2x - 6,;, 2 < x ≤ 3),([x - 3] + 9,;, 3 < x ≤ 5),(2x + 1,;, x > 5):}`

where [t] is the greatest integer less than or equal to t. Let m be the number of points where f is not differentiable and I = `int_-2^2f(x)dx`. Then the ordered pair (m, I) is equal to `underlinebb((4, 27/4)`.

Explanation:

Given: f: R → R

f(x) = `{{:(max_(t ≤ x){t^3 - 3t}, x ≤ 2),(x^2 + 2x - 6, 2 < x ≤ 3),([x - 3] + 9, 3 < x ≤ 5),(2x + 1, x > 5):}`

The graph of y = t³ – 3t is:

For `x ≤ -1, max_(t < x) {t^3 - 3t} = x^3 - 3x`

For `-1 < x ≤ 2, max_(t < x) {t^3 - 3t} = 2`

∴ f(x) = `{{:(x^3 - 3x, x ≤ -1),(2, -1 < x ≤ 2),(x^2 + 2x - 6, 2 < x ≤ 3),(9, 3 < x ≤ 4),(10, 4 < x < 5),(11, x = 5),(2x + 1, x > 5):}`

As we know, a function is not differentiable at sharp points and at points of discontinuity

⇒ f(x) is not differentiable at x = 2, 3, 4, 5

∴ The number of points where f(x) is not differentiable = 4

⇒ m = 4

Now, I = `int_(-2)^2f(x)dx`

⇒ I = `int_(-2)^(-1) f(x)dx + int_(-1)^2 f(x)dx`

⇒ I = `int_(-2)^(-1) (x^3 - 3x)dx + int_(-1)^2 2dx`

⇒ I = `[x^4/4 - (3x^2)/2]_(-2)^(-1) + [2x]_(-1)^2`

⇒ I = `(1/4 - 3/2 - 16/4 + 6) + (4 - (-2))`

⇒ I = `(1/4 - 3/2 + 2) + 6`

⇒ I = `(1 - 6 + 32)/4`

⇒ I = `27/4`

∴ The ordered pair (m, I) = `(4, 27/4)`