Which of the following statements are correct.

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Suppose f, g and h be three real valued functions defined on R. Let fx=2x+x,gx=132x−x and hx=fgx . Then the function hfx is continuous but not derivable in R.

If f1=3,f'1=2,f''1=4 and let f−1x=gxthen g''3 is equal to −12

If y=2sin−11−x+sin−12x1−x then its derivative is zero

Let fx=xpcos1x,x≠00,x=0 then f(x) will be differentiable at x = 0 if p > 1

answer is A.

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Detailed Solution

option 1fx=2x+x=3x x>0 =x x<0 gx=132x-x =13x x>0 =x x<0when x>0hx=fgx=fxxwhen x<0hx=fgx=fx=xin both cases hx=xhence hfx=3x x≥0 x x<0 option 3 :x=sin2θ,2θ∈0,π2y=2sin−1cosθ+sin−12sinθcosθ=−2θ+2θ=0option 4:fx=2pcos1x x≠0 0 x=0when p>1then fx is continuous and differtiablehence hfxis continuous but not differtiable option 2:given fx=,f'1=2,f''1=4,andf−1x=gx⇒g3=1since fxand gx are inverse function to each otherg'x=1f'gxand g'3=12g''2=−1f'gx2f''gx.g'xg''3=−1f'g32f''g3.g'3=−14.4.12=−12

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