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CBSE Class 12-science Answered

Verify Rolle's theorem for each of the following functions on indicated intervals : f (x) = sin x –  sin 2 x on [0, p]
Asked by Topperlearning User | 04 Aug, 2014, 04:35: PM
answered-by-expert Expert Answer
Since space the space sine space function space is space continuous space space and space differentiable space everywhere comma space therefore so space are space sin space straight x space and space sin 2 straight x. space Consequently space straight f left parenthesis straight x right parenthesis equals sin space straight x minus sin space 2 straight x space is space continous space on space left square bracket 0 comma straight pi right square bracket and space differentiable space on space left parenthesis 0 comma straight pi right parenthesis Also comma space straight f left parenthesis 0 right parenthesis equals sin space 0 minus sin space 0 equals 0 and space straight f left parenthesis straight pi right parenthesis equals sin space straight pi minus sin space 2 straight pi equals 0 therefore straight f left parenthesis 0 right parenthesis equals straight f left parenthesis straight pi right parenthesis Thus comma space straight f left parenthesis straight x right parenthesis space satisfies space all space the space conditions space of space roll apostrophe straight s space theorem space on space left square bracket 0 comma straight pi right square bracket. space Consequently space there space exists space straight c element of space left parenthesis 0 comma straight pi right parenthesis space such space that space straight f apostrophe left parenthesis straight c right parenthesis equals 0 Now space straight f left parenthesis straight x right parenthesis space equals sin space straight x minus sin space 2 straight x therefore straight f apostrophe left parenthesis straight x right parenthesis equals 0 rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals cos space straight x minus 2 cos space 2 straight x rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals cos space straight x minus 2 left parenthesis 2 cos squared straight x minus 1 right parenthesis rightwards double arrow cos space straight x minus 4 cos squared straight x plus 2 equals 0 rightwards double arrow 4 cos squared straight x minus cos space straight x minus 2 equals 0 rightwards double arrow cos space straight x equals fraction numerator minus open parentheses minus 1 close parentheses plus-or-minus square root of 1 plus 32 end root over denominator 8 end fraction cos space straight x equals fraction numerator 1 plus-or-minus square root of 33 over denominator 8 end fraction equals 0.8431 space or space minus 0.5931 straight x equals cos to the power of minus 1 end exponent left parenthesis 0.8431 right parenthesis space or space cos to the power of minus 1 end exponent left parenthesis minus 0.5931 right parenthesis straight x equals 32.5 to the power of straight o or space 126.38 to the power of straight o Thus space straight c equals 32.5 to the power of straight o & space 126.38 to the power of straight o space element of left parenthesis 0 comma straight pi right parenthesis space such space that space straight f apostrophe left parenthesis straight c right parenthesis equals 0 Hence comma space Rolle apostrophe straight s space theorem space is space verified.
Answered by | 04 Aug, 2014, 06:35: PM

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CBSE 12-science - Maths
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