Find the values of a & b such that the function defined by f(x)= {(5, if x ≤ 2) (ax + b, if 2 < x <10) (21, if x ≥ 10) is a continuous function

Solution:

The given function is 

f(x)= {(5, if x ≤ 2) (ax + b, if 2 < x <10) (21, if x ≥ 10)

It is evident that f is defined at all points of the real line.

If f is a continuous function, then f is continuous at all real numbers.

In particular, f is continuous at x = 2 and x = 10

Since f is continuous at x = 2, we obtain

lim_x→2⁻ f(x) = lim_x→2⁺ f(x) = f(2)

⇒ lim_x→2⁻ (5) = lim_x→2⁺ (ax + b) = 5

⇒ 5 = 2a + b = 5

⇒ 2a + b = 5 .…(1)

Since f is continuous at x = 10, we obtain

lim_x→10⁻ f(x) = lim_x→10⁺ f(x) = f(10)

⇒ lim_x→10⁻ (ax + b) = lim_x→10⁺ (21) = 21

⇒ 10a + b = 21

⇒ 10a + b = 21 …(2)

On subtracting equation (1) from equation (2), we obtain

8a = 16

⇒ a = 2

By putting a = 2 in equation (1), we obtain

2(2) + b = 5

⇒ 4 + b = 5

⇒ b = 1

Therefore, the values of a and b for which f is a continuous function are 2 and 1 respectively

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 30

Find the values of a & b such that the function defined by f(x)= {(5, if x ≤ 2) (ax + b, if 2 < x <10) (21, if x ≥ 10) is a continuous function

Summary:

The values of a & b such that the function defined by f(x)= {(5, if x ≤ 2) (ax + b, if 2 < x <10) (21, if x ≥ 10) is a continuous function are 2 and 1 respectively