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Let f is twice differerntiable on R such...

Let f is twice differerntiable on R such that `f (0)=1, f'(0) =0 and f''(0) =-1,` then for `a in R, lim _(xtooo) (f((a)/(sqrtx)))^(x)=`

A

`e ^(-e ^(2))`

B

`e ^((a^(2))/(4))`

C

`e ^((a^(2))/(2))`

D

`e ^(-2a ^(2))`

Text Solution

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The correct Answer is:
C
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VIKAS GUPTA (BLACK BOOK)-CONTINUITY, DIFFERENTIABILITY AND DIFFERENTIATION-EXERCISE (SUBJECTIVE TYPE PROBLEMS)
  1. Let f is twice differerntiable on R such that f (0)=1, f'(0) =0 and f'...

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  2. Let f (x)= {{:(ac (x-1)+b,,, x lt 1),( x+2,,, 1 le x le 3),(px ^(2) +q...

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  3. If y= sin (8 sin ^(-1) x ) then (1-x ^(2)) (d^(2)y)/(dx ^(2))-x (dy)/...

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  4. If y ^(2) =4ax, then (d^(2) y)/(dx ^(2))=(ka ^(2))/( y ^(3)), where k ...

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  5. The number of values of x , x I [-2,3] where f (x) =[x ^(2)] sin (pix)...

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  6. If f (x) is continous and fifferentiable in [-3,9] and f'(x) in [-2,8]...

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  7. In f (x)= [{:(cos x ^(3),, x lt 0), ( sin x ^(3) -|x ^(3)-1|,, x ge 0)...

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  8. Let f (x) =x ^(2) +ax+3 and g (x) =x+b, where F (x) =lim (xto oo) (f(x...

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  9. Let f (x)= {{:(2-x"," , -3 le x le 0),( x-2"," , 0 lt x lt 4):} Then f...

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  10. If f (x) +2 f (1-x)( =x ^(2) +2AA x in R and f (x) is a differentiable...

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  11. Let f (x)= signum (x) and g (x) =x (x ^(2) -10x+21), then the number o...

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  12. If (d^(2))/(d x ^(2))((sin ^(4)x+ sin ^(2)x+1)/(sin ^(2)x + si n x+1))...

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  13. f (x) =a cos (piy)+b, f'((1)/(2))=pi and int (1//2)^(3//2) f (x) dx =2...

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  14. Let alpha (x) = f(x) -f (2x) and beta (x) =f (x) -f (4x) and alpha '(1...

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  15. Let f (x) =-4.e ^((1-x)/(2))+ (x ^(3))/(3 ) + (x ^(2))/(2)+ x+1 and g ...

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  16. If y=e^(2 sin ^(-1)x) then |((x ^(2) -1) y ^('') +xy')/(y)| is equal t...

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  17. Let f (x)=x+ (x ^(2))/(2 )+ (x ^(3))/(3 )+ (x ^(4))/(4 ) +(x ^(5))/(5)...

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  18. In f (x)= [{:(cos x ^(2),, x lt 0), ( sin x ^(3) -|x ^(3)-1|,, x ge 0)...

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  19. Let f :R to R be a differentiable function satisfying: f (xy) =(f(x)...

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  20. For the curve sinx+siny=1 lying in first quadrant. If underset(xrarr0...

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  21. Let f (x) = x tan ^(-1) (x^(2)) + x^(4) Let f ^(k) (x) denotes k ^(th)...

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