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Ex 5.1, 17 - Find relationship a, b so that f(x) is continuous - Ex 5.1

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Ex 5.1, 17 Find the relationship between a and b so that the function f defined by 𝑓 𝑥﷯= 𝑎𝑥+1, 𝑖𝑓 𝑥≤3﷮&𝑏𝑥+3, 𝑖𝑓 𝑥>3﷯﷯ is continuous at x = 3. Given 𝑓 𝑥﷯= 𝑎𝑥+1, 𝑖𝑓 𝑥≤3﷮&𝑏𝑥+3, 𝑖𝑓 𝑥>3﷯﷯ Given function is continuous at x = 3 f is continuous at x = 3 if L.H.L = R.H.L = 𝑓 3﷯ i.e. if lim﷮x→ 3﷮−﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 3﷮+﷯﷯ 𝑓 𝑥﷯ = 𝑓 1﷯ And 𝑓 𝑥﷯=ax+1 So, 𝑓 3﷯=3a +1 Now, lim﷮x→ 3﷮−﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 3﷮+﷯﷯ 𝑓 𝑥﷯ = 𝑓 1﷯ 3𝑎 + 1 = 3𝑏 + 3 = 3𝑎 + 1 ⇒ 3𝑎 + 1 = 3𝑏 + 3 ⇒ 3𝑎−3b=3−1 ⇒ 3𝑎 −3𝑏=2 ⇒ 3 𝑎−𝑏﷯=2 ⇒ 𝑎−𝑏= 2﷮3﷯ ⇒ 𝑎=𝑏+ 2﷮3﷯ Thus , for any arbitrary value of b. We can find value of a.

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