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Let f and g be real-valued functions. If _ x 0 2 f(x)-g(x) [f(x)+7)]^ 2 / 3 = 7 4 , _ x 0 f(x)=1 and _ x 0 g(x)= , then h(x)= cases ( x), & 0 x 10 (2 x), & 10 < x 5 cases

Options

  1. Acontinuous at x= 10 only
  2. Bdiscontinuous on [0, 5 ]
  3. Cdiscontinuous at x= 10
  4. Dcontinuous on [0, 5 ]

Correct answer

D. continuous on [0, 5 ]

Step-by-step solution

It is given that _ x 0 f(x)=1, _ x 0 g(x)= and _ x 0 2 f(x)-g(x) (f(x)+7)^ 2 / 3 = 7 4 2- (8)^ 2 / 3 = 7 4 =-5 For h(x)= cases ( x), & 0 x 10 (2 x), & 10 < x 5 cases =-5 , so ( x) is continuous for x [0, 10 ) and (2 x) is also continuous for x ( 10 , 5 ] Now, at x= / 10 aligned LHL ( at x= 10 ) & = _ h 0 (-5 ( 10 -h ) ) & = (- 2 )=-1 aligned and f ( 10 )= (- 5 10 )= (- 2 )=-1 and RHL ( . at .x= 10 )= _ h 0 (-10 ( 10 +h ) )= _ h 0 ( +10 h)= =-1 The function h(x) is continuous on [0, 5 ] . Hence, option (d) is correc

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