Verify the truth of Rolle's Theorem for the function from Mathe
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    Continuity and Differentiability

    Multiple Choice QuestionsShort Answer Type

    591. Verify Roll's Theorem for the function :f(x) = x (x - 3)2 in 0 ≤ x ≤ 3
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    592. Verify Roll's Theorem for the function :f(x) = (x2 - 1) (x - 2) in [-1, 2]
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    593. Verify Roll�s Theorem for the function :f{x) = e1 - x2 in [-1, 1]
    82 Views
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    594. Verify Roll�s Theorem for the function :f(x) = log (x2 + 2) - log 3 in [-1, 1]
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    595. Verify the truth of Rolle's Theorem for the function
    straight f left parenthesis straight x right parenthesis equals space sin space 3 straight x comma space straight x element of open square brackets 0 comma space straight pi over 3 close square brackets

    Here space straight f left parenthesis straight x right parenthesis equals sin space 3 straight x left parenthesis straight i right parenthesis space Since space sine space curve space is space continuous space therefore space straight f left parenthesis straight x right parenthesis space is space continuous space function space in space open square brackets 0 comma space straight pi over 3 close square brackets left parenthesis ii right parenthesis space straight f apostrophe left parenthesis straight x right parenthesis equals 3 space cos space 3 straight x comma space which space exists space in space open parentheses 0 comma space straight pi over 3 close parentheses therefore space straight f left parenthesis straight x right parenthesis space is space derivable space in space open parentheses 0 comma space straight pi over 3 close parentheses left parenthesis iii right parenthesis straight f left parenthesis 0 right parenthesis equals sin space 0 equals 0 comma space straight f open parentheses straight pi over 3 close parentheses equals sin space straight pi equals 0 therefore space straight f left parenthesis 0 right parenthesis equals straight f open parentheses straight pi over 3 close parentheses therefore space straight f left parenthesis straight x right parenthesis space satisfies space all space the space conditions space of space Rolle apostrophe straight s space theorem. space therefore space there space must space exist space at space least space one space value space straight c space of space straight x space such space that space straight f apostrophe left parenthesis straight c right parenthesis space equals space 0 space where straight c element of space open parentheses 0 comma space straight pi over 3 close parentheses. Now space straight f apostrophe left parenthesis straight c right parenthesis equals 3 space cos space 3 straight c therefore space straight f apostrophe left parenthesis straight c right parenthesis equals 0 space gives space us space 3 space cos space 3 straight c equals 0 space straight i. straight i. comma space cos space 3 space straight c equals 0 therefore space space space 3 space straight c equals straight pi over 2 comma space fraction numerator 3 straight pi over denominator 2 end fraction comma space fraction numerator 5 straight pi over denominator 2 end fraction comma... therefore space space space space space space straight c equals straight pi over 6 comma space fraction numerator 3 straight pi over denominator 2 end fraction comma space fraction numerator 5 straight pi over denominator 6 end fraction comma... space Now space straight c equals straight pi over 6 space lies space in space open parentheses 0 comma space straight pi over 3 close parentheses
    ∴ Rolle's Theorem is verified.
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    596. Verify Rolle's Theorem for the function f(x) = sin2 x in [0, straight pi].
    95 Views
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    597. Verify Rolle's Theorem for the function straight f left parenthesis straight x right parenthesis space equals space cos squared space straight x space in space left square bracket 0 comma space straight pi right square bracket
    75 Views
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    598. Verify Roll's Theorem for the function :straight f left parenthesis straight x right parenthesis equals space sin space straight x minus 1 space in space open square brackets straight pi over 2 comma space fraction numerator 5 straight pi over denominator 2 end fraction close square brackets
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    599. Verify Roll's Theorem for the function : straight f left parenthesis straight x right parenthesis equals space sin space 2 space straight x space or space cos space 2 open parentheses straight x minus straight pi over 4 close parentheses space in space open square brackets 0 comma space straight pi over 2 close square brackets
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    600. Verify Roll's Theorem for the function : straight f left parenthesis straight x right parenthesis equals cos space straight x plus sin space straight x space in space left square bracket 0 comma space 2 straight pi right square bracket
    99 Views
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