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 | 4th Aug, 2014, 04:35: PM

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  | 4th Aug, 2014, 06:35: PM