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For every pair of continuous functions f...

For every pair of continuous functions `f,g:[0,1]->R` such that `max{f(x):x in [0,1]}= max{g(x):x in[0,1]}` then which are the correct statements

A

`[f(c)]^(2) + 3f(c) = [g(c)]^(2) + 3 g(c)"for some c" in [0, 1]1`

B

`[f(c)]^(2) + f(c) = [g(c)]^(2) + 3g(c)"for some c" in [0, 1]`

C

`[f(c)]^(2) + 3f(c) = [g(c)]^(2) + g(c)"for some c" in [0, 1]`

D

`[f(c)]^(2) = [g(c)]^(2)"for some c" in [0, 1]`

Text Solution

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The correct Answer is:
A, D
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Knowledge Check

  • For every pair of continuous functions f,g:[0,1] to R such that max {f(x) : x in [0,1]} =" max " {g(x) :x in [0,1]} , the correct statement(s) is (are).

    A
    `(f( c))^(2) + 3f( c) = (g(c ))^(2) + 3g( c)` for some `c in [0,1]`
    B
    `(f( c))^(2) + f( c) = (g( c))^(2) + 3g (c )` for some `c in [0,1]`
    C
    `(f(c ))^(2) + 3f (c ) = (g( c))^(2) + g( c)` for some `c in [0,1]`
    D
    `(f( c))^(2) = (g( c))^(2)` for some `c in [0,1]`

  • Let f and g be continuous functions on [ 0 , a] such that f(x)=f(x)=f(a-x)andg(x)+g(a-x)=4 , then int_(0)^(a) f(x)g(x) dx is equal to

    A
    `4int_(0)^(a)f(x)`dx
    B
    `int_(0)^(a)f(x)`dx
    C
    `2int_(0)^(a)f(x)`dx
    D
    `-3int_(0)^(a)f(x)`dx

  • If f and g are continuous functions in [0,1] satisfying f(x)=f(a-x)" and "g(x)+g(a-x)=a , then int_0^af(x).g(x)dx is equal to

    A
    `a/2`
    B
    `a/2int_0^af(x)dx`
    C
    `int_0^af(x)dx`
    D
    `aint_0^af(x)dx`

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