Limit Continuity and Differentiability 7 Question 8

8. Let f(x)=max|x|,x2,|x|2 82|x|,2<|x|4

Let S be the set of points in the interval (4,4) at which f is not differentiable. Then, S

(2019 Main, 10 Jan I)

(a) equals 2,1,0,1,2

(b) equals 2,2

(c) is an empty set

(d) equals 2,1,1,2

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Answer:

Correct Answer: 8. (d)

Solution:

  1. We have,

f(x)=x3+x2f(1)+xf(2)+f(3)f(x)=3x2+2xf(1)+f(2)f(x)=6x+2f(1)f(x)=6f(3)=6

Putting x=1 in Eq. (i), we get

f(1)=3+2f(1)+f(2)

and putting x=2 in Eq. (ii), we get

f(2)=12+2f(1)

From Eqs. (iv) and (v), we get

f(1)=3+2f(1)+(12+2f(1))3f(1)=15f(1)=5f(2)=12+2(5)=2 [using Eq. (v)] f(x)=x3+x2f(1)+xf(2)+f(3)f(x)=x35x2+2x+6f(2)=235(2)2+2(2)+6=820+4+6=2



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