A function f(x) is defined as follows for real xfx = 1&

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 Multiple Choice QuestionsShort Answer Type

71.

If x = sint, y = sin2t, prove that

1 - x2d2ydx2 - xdydx + 4y = 0


72.

If f is differentiable at x = a, find the value of

limxax2fa - a2fxx - a


 Multiple Choice QuestionsMultiple Choice Questions

73.

Rolle's theorem is not applicable to the function f(x) = x for - 2 x  2 because

  • f is continuous for - 2 x  2

  • f is not derivable for x = 0

  • f(- 2) = f(2)

  • f is not a constant function


74.

The function f(x) which satisfies

fx = f- x = f'(x)x is given by

  • f(x) = 12ex2

  • f(x) = 12e- x2

  • f(x) = x2ex2/2

  • f(x) = ex2/2


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75.

A function f(x) is defined as follows for real x

fx = 1 - x2,      for x < 10,               for x = 11 - x2,      for x > 1

Then,

  • f (x) is not continuous at x = 1

  • f (x) is continuous but not differentiable at x = 1

  • f(x) is both continuous and differentiable at x = 1

  • None of the above


A.

f (x) is not continuous at x = 1

Since, fx = 1 - x2,      for x < 10,               for x = 11 - x2,      for x > 1     LHL = limx1-fx = limh01 - 1 - h2                = 0and  RHL = limx1+fx = limh01 + 1 + h2 = 0               = 2Also, f(1) = 0 RHL  LHL = f(1)

Thus, f(x) is not continuous at x = 1. 


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76.

Select the correct statement from (a), (b), (c), (d). The function f(x) = xe1 - x

  • strictly increases in the interval 12, 2

  • increases in the interval (0, )

  • decreases in the interval (0, 2)

  • strictly decreases in the interval (1, )


77.

The function f(x) = eax + e- ax, a > 0 is monotonically increasing for

  • - 1 < x < 1

  • x < - 1

  • x > - 1

  • x > 0


78.

If y = ax . b2x - 1, then d2ydx2 is

  • y2logab2

  • ylogab2

  • y2

  • ylogab22


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79.

Let f(x) = ex, g(x) = sin-1x and h(x) = f[g(x)], then h'(x)hx is equal to

  • esin-1x

  • 11 - x2

  • sin-1x

  • 11 - x2


80.

If fx = logx, then

  • f(x) is continuous and differentiable for all x in its domain

  • f(x) is continuous for all x in its domain but not differentiable at x = ± 1

  • f(x) is neither continuous nor differentiable at x = ± 1

  • None of the above


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