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Asked by mnsdewari | 3rd Apr, 2020, 06:52: AM

Expert Answer:

Q: Let S = {t belongs to R: f(x) = |x-pi|(e^|x| - 1) sin|x| is not differentiable at t}, then the set S is equal to
Solution:
S equals open curly brackets t element of R space colon space f left parenthesis x right parenthesis space equals space vertical line x minus straight pi vertical line open parentheses straight e to the power of vertical line straight x vertical line end exponent minus 1 close parentheses sin to the power of vertical line straight x vertical line end exponent close curly brackets
f open parentheses x close parentheses equals open curly brackets table attributes columnalign left end attributes row cell left parenthesis x minus straight pi right parenthesis open parentheses straight e to the power of negative x end exponent minus 1 close parentheses sin space straight x space space semicolon space space straight x less than 0 end cell row cell negative left parenthesis x minus straight pi right parenthesis open parentheses straight e to the power of x minus 1 close parentheses sin space straight x space space semicolon space space 0 less or equal than straight x less than straight pi end cell row cell left parenthesis x minus straight pi right parenthesis open parentheses straight e to the power of x minus 1 close parentheses sin space straight x space space semicolon space space straight x greater or equal than straight pi end cell end table close
f apostrophe open parentheses x close parentheses equals open curly brackets table attributes columnalign left end attributes row cell left parenthesis x minus straight pi right parenthesis open parentheses straight e to the power of negative x end exponent minus 1 close parentheses space cos space straight x plus left parenthesis straight e to the power of negative straight x end exponent minus 1 right parenthesis space sin space straight x minus left parenthesis straight x minus straight pi right parenthesis sin space straight x space straight e to the power of negative straight x end exponent space space semicolon space space straight x less than 0 end cell row cell negative left parenthesis x minus straight pi right parenthesis open parentheses straight e to the power of x minus 1 close parentheses cos space straight x minus open parentheses straight e to the power of straight x minus 1 close parentheses sin space straight x minus left parenthesis straight x minus straight pi right parenthesis open parentheses straight e to the power of straight x close parentheses sin space straight x space semicolon space space 0 less or equal than straight x less than straight pi end cell row cell left parenthesis x minus straight pi right parenthesis open parentheses straight e to the power of x minus 1 close parentheses cos space straight x plus open parentheses straight e to the power of straight x minus 1 close parentheses sin space straight x plus left parenthesis straight x minus straight pi right parenthesis open parentheses straight e to the power of straight x close parentheses sin space straight x space space semicolon space space straight x greater or equal than straight pi end cell end table close

T o space c h e c k space t h e space d i f f e r e n t i a b i l i t y space o f space f space a t space straight pi space and space 0
At space straight x equals 0 comma
RHD equals straight f apostrophe left parenthesis 0 to the power of plus right parenthesis equals limit as straight x rightwards arrow 0 of open square brackets negative left parenthesis straight x minus straight pi right parenthesis open parentheses straight e to the power of straight x minus 1 close parentheses cos space straight x minus open parentheses straight e to the power of straight x minus 1 close parentheses sin space straight x minus left parenthesis straight x minus straight pi right parenthesis open parentheses straight e to the power of straight x close parentheses sin space straight x close square brackets equals 0
Similarly comma space LHD equals straight f apostrophe left parenthesis 0 to the power of minus right parenthesis equals limit as straight x rightwards arrow 0 of open square brackets left parenthesis straight x minus straight pi right parenthesis open parentheses straight e to the power of negative straight x end exponent minus 1 close parentheses space cos space straight x plus left parenthesis straight e to the power of negative straight x end exponent minus 1 right parenthesis space sin space straight x minus left parenthesis straight x minus straight pi right parenthesis sin space straight x space straight e to the power of negative straight x end exponent close square brackets equals 0
rightwards double arrow straight f apostrophe left parenthesis 0 to the power of plus right parenthesis equals straight f apostrophe left parenthesis 0 to the power of minus right parenthesis

At space straight x equals straight pi comma
LHD equals straight f apostrophe left parenthesis straight pi to the power of plus right parenthesis equals limit as straight x rightwards arrow straight pi of open square brackets negative left parenthesis straight x minus straight pi right parenthesis open parentheses straight e to the power of straight x minus 1 close parentheses cos space straight x minus open parentheses straight e to the power of straight x minus 1 close parentheses sin space straight x minus left parenthesis straight x minus straight pi right parenthesis open parentheses straight e to the power of straight x close parentheses sin space straight x close square brackets equals 0
Similarly comma space RHD equals straight f apostrophe left parenthesis 0 to the power of minus right parenthesis equals limit as straight x rightwards arrow straight pi of open square brackets left parenthesis straight x minus straight pi right parenthesis open parentheses straight e to the power of straight x minus 1 close parentheses cos space straight x plus open parentheses straight e to the power of straight x minus 1 close parentheses sin space straight x plus left parenthesis straight x minus straight pi right parenthesis open parentheses straight e to the power of straight x close parentheses sin space straight x close square brackets equals 0
rightwards double arrow straight f apostrophe left parenthesis straight pi to the power of plus right parenthesis equals straight f apostrophe left parenthesis straight pi to the power of minus right parenthesis

Therefore comma space straight f space is space differentiable space at space straight x equals 0 space and space straight x equals straight pi.
Hence comma space the space set space straight S equals empty set

Answered by Renu Varma | 6th Apr, 2020, 02:35: PM

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