           1. Chapter 5 Class 12 Continuity and Differentiability
2. Serial order wise
3. Miscellaneous

Transcript

Misc 21 Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer. Consider the function 𝑓 𝑥﷯= 𝑥﷯+ 𝑥−1﷯ 𝑓 is continuous every where , but it is not differentiable at 𝑥 = 0 & 𝑥 = 1 . 𝑓 𝑥﷯= −𝑥− 𝑥−1﷯ 𝑥≤0﷮𝑥− 𝑥−1﷯ 0<𝑥<1﷮𝑥+ 𝑥−1﷯ 𝑥≥1﷯﷯ = −2𝑥+1 𝑥≤0﷮ 1 0<𝑥<1﷮2𝑥−1 𝑥≥1﷯﷯ Checking continuity Case 1 :- At 𝑥<0 𝑓 𝑥﷯=−2𝑥+1 𝑓 𝑥﷯ is polynomial ⇒ 𝑓 𝑥﷯ is continuous Case 2 :- At 𝑥>1 𝑓 𝑥﷯=2𝑥−1 𝑓 𝑥﷯ is polynomial ⇒ 𝑓 𝑥﷯ is continuous Case 3 :- At 0<𝑥<1 𝑓 𝑥﷯=1 𝑓 𝑥﷯ is constant, ⇒ 𝑓 𝑥﷯ is continuous Case 4 :- At 𝑥=0 𝑓 𝑥﷯ = −2𝑥+1 𝑥≤0﷮ 1 0<𝑥<1﷮2𝑥−1 𝑥≥1﷯﷯ A function is continuous at 𝑥=0 if LHL = RHL = 𝑓 0﷯ i.e. lim﷮𝑥 → 0﷮−﷯ ﷯ 𝑓 𝑥﷯ = lim﷮𝑥 → 0﷮+﷯ ﷯ 𝑓 𝑥﷯ = 𝑓 0﷯ & 𝑓 𝑥﷯= −2𝑥+1 𝑓 0﷯= −2 0﷯+1= 1 Hence LHL = RHL = f (0) ⇒ 𝑓 is continuous . Case 5 :- At 𝑥=1 A function is continuous at 𝑥=1 if LHL = RHL = 𝑓 1﷯ i.e. lim﷮𝑥 → 1﷮−﷯ ﷯ 𝑓 𝑥﷯ = lim﷮𝑥 → 1﷮+﷯ ﷯ 𝑓 𝑥﷯ = 𝑓 1﷯ & 𝑓 𝑥﷯=2𝑥−1 𝑓 1﷯=2 1﷯−1=2−1 =1 Hence LHL = RHL = 𝑓 1﷯ ⇒ 𝑓 is continuous at 𝑥=1 Thus 𝑓 𝑥﷯= 𝑥﷯+ 𝑥−1﷯ is continuous for all value of 𝑥 . To check differentiability Case 1 :- At 𝑥<1 𝑓 𝑥﷯=−2𝑥+1 𝑓 𝑥﷯ is polynomial ⇒ 𝑓 𝑥﷯ is differentiable Case 2 :- At 𝑥>1 𝑓 𝑥﷯=2𝑥−1 𝑓 𝑥﷯ is polynomial ⇒ 𝑓 𝑥﷯ is differentiable Case 3 :- At 0<𝑥<1 𝑓 𝑥﷯=1 𝑓 𝑥﷯ is constant ⇒ 𝑓 𝑥﷯ is differentiable Case 4 :- At 𝑥=0 We know that 𝑓 is differentiate at 𝑥 = 0 If L.H.D = R.H.D = 𝑓﷮′﷯ 0﷯ i.e., lim﷮ℎ → 0﷮−﷯ ﷯ 𝑓 0﷯ − 𝑓 0 − ℎ﷯﷮ℎ﷯ = lim﷮ℎ → 0﷮+﷯ ﷯ 𝑓 0 + ℎ﷯ − 𝑓 0﷯﷮ℎ﷯ = 𝑓﷮′﷯ 𝑐﷯

Miscellaneous 