Verify Roll's Theorem for the function :  from Mathematics Con

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 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]
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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
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596. Verify Rolle's Theorem for the function f(x) = sin2 x in [0, straight pi].
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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
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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


space Here space straight f left parenthesis straight x right parenthesis space equals space cos space straight x space plus space sin space 2 straight x space space
left parenthesis straight a right parenthesis space Since space cos space straight x space and space sin space 2 straight x space are space both space continuous space in space left square bracket 0 comma space 2 space straight pi right square bracket space space
therefore space straight f left parenthesis straight x right parenthesis space equals space cos space straight x space plus space sin space straight x space is space continuous space in space left square bracket 0 comma space 2 space straight pi right square bracket space space
left parenthesis straight b right parenthesis space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus space sin space straight x space plus space cos space straight x comma space which space exists space in space left parenthesis 0 comma space 2 space straight pi right parenthesis space space
therefore space straight f left parenthesis straight x right parenthesis space is space derivable space in space left parenthesis 0 comma space 2 space straight pi right parenthesis space space
left parenthesis straight c space right parenthesis space straight f left parenthesis 0 right parenthesis space equals space cos space 0 space plus space sin space 0 space equals space 1 space plus space 0 space equals space 1 space
space straight f left parenthesis 2 space straight pi right parenthesis space equals space cos space 2 space straight pi space plus space sin space 2 space straight pi space equals space 1 space plus space 0 space equals space 1 space space
therefore space straight f left parenthesis 0 right parenthesis space equals space straight f left parenthesis 2 space straight pi right parenthesis space space
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 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 space left parenthesis straight c right parenthesis space equals space 0 space where space 0 space less than space straight c space less than space 2 space straight pi space space
Now space straight f apostrophe left parenthesis straight c right parenthesis space equals space 0 space gives space minus space sin space straight c space plus space cos space straight c space equals space 0 space rightwards double arrow space tan space straight c space equals space 0 space space
therefore space straight c space equals space straight pi space element of space left parenthesis 0 comma space 2 space straight pi right parenthesis space space
therefore space Rolle apostrophe straight s space Theorem space is space verified.
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