Question

Define `f(x) = {(x^2 + bx + c",", x < 1), (             x",", x ≥ 1):}`. If f(x) is differentiable at x = 1, then (b – c) is equal to ______.

पर्याय

• −2

• 0

• 1

• 2

Answer 1

Define `f(x) = {(x^2 + bx + c",", x < 1), (             x",", x ≥ 1):}`. If f(x) is differentiable at x = 1, then (b – c) is equal to −2.

Explanation:

`f'(x) = {(2x + b",", x < 1), (      1",", x ≥ 1):}`

Since, f(x) is differentiable at x = 1.

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

`lim_(x→1^-) (2x + b) = lim_(x→1^+) 1`

2 + b = 1

= b = −1

As form is differentiable at x = 1 So, it will be continuous at x = 1 also.

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

= `lim_(x→1^-) x^2 + bx = c = lim_(x→1^+) x = 1`

= 1 + b + c = 1

= 1 − 1 + c = 1 ⇒ c = 1

Hence, b − с = −1 −1 = −2