Answer :

As the function f(x) is differentiable at x = 1 hence it is also continuous at x = 1 because differentiable function is also continuous.

As f(x) is continuous at x = 1

left hand limit of 1 = right hand limit of 1 = f(1)

As f(x) = 2x + 1 for x ≥ 1

⇒ f(1) = 2(1) + 1 = 3 …(p)

Now left-hand limit

For x < 1 f(x) = ax^{2} + b

⇒ left hand limit = a (1)^{2} + b

⇒ left hand limit = a + b …(q)

Now right-hand limit

For x > 1 f(x) = 2x + 1

⇒ right hand limit = 2(1) + 1

⇒ right hand limit = 3 …(r)

Equating (p), (q) and (r) as f(x) is continuous

⇒ a + b = 3 …(i)

As f(x) is differentiable at x = 1 hence the left-hand derivative and the right hand derivative should be equal

Let us find the left-hand derivative

We have to check differentiability at 1

For x = 1 f(x) = 2x + 1 and left-hand limit refers to the left hand side of 1 which means less than 1, for x < 1 f(x) = ax^{2} + b

Using (i)

⇒ left hand derivative = 2a …(j)

Now right hand derivative

We have to check differentiability at 1

For x = 1 f(x) = 2x + 1 and right hand limit refers to the right hand side of 1 which means greater than 1, for x > 1 f(x) = 2x + 1

⇒ right hand derivative = 2 …(k)

Equate (j) and (k) as f(x) is differentiable

⇒ 2a = 2

⇒ a = 1

Pute a = 1 in (i)

⇒ 1 + b = 3

⇒ b = 2

Hence if f(x) is differentiable at x = 1 then a = 1 and b = 2

**OR**

For f(x) to be continuous at x = 0 left hand limit of 0 = right hand limit of 0 = f(0)

⇒ f(0) = 2 …given …(i)

Now left-hand limit

⇒left hand limit

For x < 0

⇒left hand limit

Divide numerator and denominator by x

Now for multiply divide by (a + 1)

We know that

Now let us find the right-hand limit

For x > 0

⇒right hand limit

Multiply divide by sin bx

We know that

⇒ right hand limit = 2 …(iii)

Note: Verify using L'Hopital's rule that

Equate (i), (ii) and (iii) for f(x) to be continuous at x = 0

⇒ a + 1 = 1

⇒ a = 1 - 1

⇒ a = 0

As there is no equation for b.

So, b can be any number other than zero because the continuity of f(x) is not dependent on b.

(Why can’t b = 0? Observe that when x > 0 , if here b = 0 then f(x) would be undefined because denominator will become 0 hence b ≠ 0)

Hence for f(x) to be continuous a = 0 and b can be any number but b ≠ 0.

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