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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

51.

ધારો કે f : R →R, f(x+y) = f(x) + f(y); ∀ x, y ∈ R. જો f(x) એ x = 0 આગળ વિકલનીય હોય, તો

f એ R પર સતત નથી.
f(x) એ અમુક બિંદું સિવાય વિકલનીય છે.
f(x) એ (a, b)જ્યાં a < 0 < b અંતરાલમાં વિકલનીય હોય.
f(x) = અચળ

52.

$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold g bold apostrophe bold left parenthesis bold x bold right parenthesis bold space fraction numerator bold e to the power of begin display style bold a over bold x end style end exponent bold minus bold e to the power of bold minus begin display style bold a over bold x end style end exponent over denominator bold g to the power of begin display style bold a over bold x end style end exponent bold space bold plus bold space bold e to the power of bold minus begin display style bold a over bold x end style end exponent end fraction bold space$ જ્યાં g' એ વિધેય g નું વિકલિત છે તથા સતત વિધેય છે અને a > 0. જો $bold lim with bold x bold rightwards arrow bold 0 below bold space bold f bold left parenthesis bold x bold right parenthesis$ નું અસ્તિત્વ હય તો

g(x) = a₀ + a₁x + a₂x² + ..... + a_nx^-n જ્યાં a_i ∈ R, i = 1,2 ....n તથા a₁ # 0 a_n # 0
g(x) = log (1 + x)
g એવું વિધેય હોય જ્યાં g'(0) = 0
g(x0 = sin x

53. નીચેનામાંથી કયું વિધેય અંતરાલ [-1, 1] માં રોલના પ્રમેયની શરતોનું પાલન કરે છે.

f(x) = [x] + [-x]
f(x) = |x| [x]
f(x) = |x| - |sin x|
$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell fraction numerator bold sin bold space bold x over denominator bold x end fraction bold comma bold space bold x bold # bold 0 bold space end cell row cell bold 0 bold space bold comma bold space bold x bold space bold equals bold space bold 0 bold space end cell end table close$

f(x) = |x| - |sin x|

Tips: -

$bold left parenthesis bold a bold right parenthesis bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold vertical line bold x bold vertical line bold space bold left square bracket bold x bold right square bracket bold space bold equals bold space open curly brackets table row cell bold x bold comma end cell cell bold minus bold 1 bold less-than or slanted equal to bold x bold less than bold 0 end cell row cell bold 0 bold comma end cell cell bold 0 bold less or equal than bold x bold less than bold 1 end cell row cell bold 1 bold comma end cell cell bold x bold equals bold 1 end cell end table close bold therefore bold space bold f bold space bold એ bold space bold x bold space bold equals bold space bold 0 bold space bold space bold મ ા ટ ે bold space bold વ િ કલન ી ય bold space bold નથ ી bold. bold space bold left parenthesis bold b bold right parenthesis bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold left square bracket bold x bold right square bracket bold space bold plus bold space bold left square bracket bold minus bold x bold right square bracket bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell bold 0 bold comma end cell row cell bold minus bold 1 bold comma end cell end table close table row cell bold x bold element of bold Z end cell row cell bold x bold not an element of bold Z end cell end table bold space bold therefore bold space bold f bold space bold એ bold space bold x bold space bold equals bold space bold 0 bold space bold આગળ bold space bold સતત bold space bold નથ ી bold. bold space bold left parenthesis bold c bold right parenthesis bold space bold lim with bold x bold rightwards arrow bold 0 below bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold lim with bold x bold rightwards arrow bold 0 below bold space bold sinx over bold x bold space bold equals bold space bold 1 bold space bold અન ે bold space bold f bold left parenthesis bold 0 bold right parenthesis bold space bold equals bold space bold 0 bold space bold therefore bold space bold space bold f bold space bold એ bold space bold x bold space bold equals bold space bold 0 bold space bold આગળ bold space bold સતત bold space bold નથ ી bold. bold space bold left parenthesis bold d bold right parenthesis bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell bold x bold space bold minus bold space bold sinx end cell row cell bold minus bold x bold space bold plus bold space bold sinx end cell end table close table row cell bold 0 bold less or equal than bold space bold x bold space bold less or equal than bold space bold 1 end cell row cell bold minus bold 1 bold space bold less or equal than bold space bold x bold space bold less or equal than bold 0 end cell end table$

∴ f એ [-1, 1] માં સતત છે. તેમજ (-1, 1)માં વિકલનીય છે.

વળી, f(-1) = f(1)

આમ, રોલના પ્રમેયની શરતોનું પાલન થાય છે.