# Find ‘a’ and ‘b’, 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) = ax2 + 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) = ax2 + 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.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation view all courses RELATED QUESTIONS :

Find which of theMathematics - Exemplar

Discuss the contiRD Sharma - Volume 1

Find which of theMathematics - Exemplar

Find which of theMathematics - Exemplar

If <iMathematics - Exemplar

<img width=Mathematics - Exemplar

Find the value ofMathematics - Exemplar

Discuss the contiRD Sharma - Volume 1

Discuss the contiRD Sharma - Volume 1

Find the value ofMathematics - Exemplar