NCERT Exemplar - MCQs

Chapter 5 Class 12 Continuity and Differentiability
Serial order wise

## (D) none of these.

This question is similar to Ex 5.1, 32 - Chapter 5 Class 12 and Ex 5.2, 9 - Chapter 5 Class 12 - Continuity and Differentiability

### Transcript

Question 15 The function f (x) = ๐^(|๐ฅ|) is (A) continuous everywhere but not differentiable at x = 0 (B) continuous and differentiable everywhere (C) not continuous at x = 0 (D) none of these. f(๐ฅ) = ๐^(|๐ฅ|) We need to check continuity and differentiability of f(๐ฅ) Continuity of f(๐) Let ๐(๐)=๐^๐ & ๐(๐)=|๐| Then, ๐๐๐(๐)=๐(โ(๐ฅ)) =๐(|๐ฅ|) =๐^|๐ฅ| =๐(๐) โด ๐(๐ฅ)=๐๐โ(๐ฅ) We know that, ๐(๐)=|๐| is continuous as it is modulus function ๐(๐)=๐^๐ฅ is continuous as it is an exponential function Hence, g(๐ฅ) & h(๐ฅ) both are continuous And If two functions g(๐ฅ) & h(๐ฅ) are continuous then their composition ๐๐โ(๐ฅ) is also continuous โด ๐(๐) is continuous Differentiability of ๐(๐) ๐(๐ฅ)=๐^(|๐ฅ|) ๐(๐ฅ)={โ 8(๐^๐ฅ, ๐ฅโฅ0@๐^(โ๐ฅ), ๐ฅ<0)โค Now, ๐(๐ฅ) is differentiable at ๐ฅ=0, if LHD = RHD (๐๐๐ )โฌ(๐กโ๐) (๐(๐) โ ๐(๐ โ ๐))/๐ = (๐๐๐)โฌ(hโ0) (๐(0) โ ๐(0 โ โ))/โ = (๐๐๐)โฌ(hโ0) (๐^(|0|)โ ๐^(|0 โโ|))/โ = (๐๐๐)โฌ(hโ0) (๐^(|0|)โ ๐^(| โโ|))/โ = (๐๐๐)โฌ(hโ0) (๐^0 โ ๐^โ)/โ = (๐๐๐)โฌ(hโ0) (1 โ ๐^โ)/โ = (๐๐๐)โฌ(hโ0) (โ(๐^โ โ 1))/โ Using (๐๐๐)โฌ(xโ0) (๐^๐ฅ โ 1)/๐ฅ=1 = (๐๐๐)โฌ(hโ0) โ1 = โ1 (๐๐๐ )โฌ(๐กโ๐) (๐(๐ + ๐) โ ๐(๐ ))/๐ = (๐๐๐)โฌ(hโ0) (๐(0 + โ) โ ๐(0))/โ = (๐๐๐)โฌ(hโ0) (๐^(|0 + โ|) โ๐^(|0|))/โ = (๐๐๐)โฌ(hโ0) (๐^(|โ|) โ๐^0)/โ = (๐๐๐)โฌ(hโ0) (๐^โ โ 1)/โ Using (๐๐๐)โฌ(xโ0) (๐^๐ฅ โ 1)/๐ฅ=1 = (๐๐๐)โฌ(hโ0) 1 = ๐ Since, LHD โ  RHD โด ๐(๐ฅ) is not differentiable at ๐ฅ=0 Thus, ๐(๐ฅ) continuous everywhere but not differentiable at x = 0 So, the correct answer is (A)