Ex 5.1, 30 - Find a and b such that f(x) = {5, ax + b, 21

Ex 5.1, 30 Find the values of a and 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 𝑓 𝑥﷯= 5, 𝑖𝑓 𝑥≤2﷮𝑎𝑥+𝑏, 𝑖𝑓 2<𝑥<10﷮21, 𝑖𝑓 𝑥≥10﷯﷯ At x = 2 A function is continuous at x = 2 if L.H.L = R.H.L = 𝑓(2) i.e. lim﷮x→ 2﷮−﷯﷯ 𝑓 𝑥﷯= lim﷮x→ 2﷮+﷯﷯ 𝑓 𝑥﷯= 𝑓 2﷯ Since, LHL = RHL 2a + b = 5 At x = 10 𝑓 is continuous at x = 10 if L.H.L = R.H.L = 𝑓 10﷯ i.e. lim﷮x→ 10﷮−﷯﷯ 𝑓 𝑥﷯= lim﷮x→ 10﷮+﷯﷯ 𝑓 𝑥﷯= 𝑓 10﷯ Since, L.H.L = R.H.L 10a + b = 21 Now our equation are 2a + b = 5 …(1) 10 a + b = 21 …(2) From (1) 2a + b = 5 b = 5 − 2a Putting value of b in (2) 10 𝑎+ 5−2𝑎﷯ = 21 10 𝑎+5−2𝑎 = 21 8𝑎 = 21−5 8𝑎 = 16 𝑎 = 16﷮8﷯ 𝑎 = 2 Putting value of a in (1) 2𝑎+𝑏=5 2 2﷯+𝑏=5 4+𝑏=5 𝑏=5−4 𝑏=1 Hence, a = 2 & b = 1 .

Ex 5.1, 30 - Chapter 5 Class 12 Continuity and Differentiability