Question

Let a function f: R→R be defined as

f(x) = `{(sinx - e^x",", if x < 0),(a + [-x]",", if 0 < x < 1),(2x - b",", if x > 1):}`

where [x] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to ______.

Options

• 5

• 3

• 2

• 4

Answer 1

Let a function f: R→R be defined as

f(x) = `{(sinx - e^x",", if x < 0),(a + [-x]",", if 0 < x < 1),(2x - b",", if x > 1):}`

where [x] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to 3.

Explanation:

Since, f(x) is continuous at x = 0

So, `lim_(x→0^-)f(x)=lim_(x→0+)f(x)=f(0)`

⇒ -1 = a - 1 = -1

⇒ a = 0

Since, f(x) is continuous at x = 0

So, `lim_(x→1^-)f(x)=lim_(x→1+)f(x)=f(1)`

⇒ a - 1 = 2 - b = 2 - b

Put a = 0, so 0 - 1 = 2 - b

⇒ - 3 = - b

⇒ b = 3

So, the value of a + b = 0 + 3 = 3