1.

Column -1 : real valued function, Column -2: continuity of the function, Column - 3: differentiability of the function, Match the following Column(s) (Whre [.] denotes the greatest integer function and {.} fractional at function) Which of the following combination combination is correct?

Answer»

(I)(i)`(R)`
(III)(ii)`R`
(IV)(iv)(P)
(I)(i)(Q)

SOLUTION :(I) `f(x)=||x-6|-|x-8||-|x^(2)-4|+3x-|x-7|^(3)` is continuous `AA x epsilon R` and not differentiable at `x=-2, 2, 6, 7` & `8`
(II) `f(x)=(x^(2)-9)|x^(2)+11x+24|+sin|x-7|+cos|x-4|+(x-1)^(3//5)sin(x-1)` is continuous
`AA x epsilonR` and not differentiable at `x=-8` & `7`
(III) `f(x)={((x+1)^(3//5)-(3pi)/2, : xlt-1),((x-1/2)cos^(-1)(4x^(3)-3x), : -1le x le 1), ((x-1)^(5//3),:1ltxlt2):}` is discontinuous at `x=-1` & 1 not differentiable at `x=-1, -1/2` & 1
(IV) `f(x)=P{sinx}{cosx}+(sin^(3)pi{x})([x]), x epsilon [-1, 2pi]`
Let `G(x)=underset("cont. at" x=I) ubrace((sinpi{x})([x]))(sin^(2)pi{x})`
`g^(')(I^(+))=g^(')(I^(-))` so differentiable at `x=I` and for `{sinx}{cos}`
Doubtful poins for non differentiabililty are `x=0, (pi)/2, pi, (3pi)/2`
`:.{sinx}.{cosx}` is discontinuous at `x=0, (pi)/2, 2pi`
So not differentiable at `x=2NPI, 2npi+(pi)/2`


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