Examine the derivability of the following functions :|x2| at x =
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    Continuity and Differentiability

    Multiple Choice QuestionsShort Answer Type

    131. Prove that the greatest integer function [x] is not differentiable at x = 1.
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    132. Prove that the function f given by f(x) = |x - 1 |, x ∈ R is not differentiable at x = 1.
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    133. Prove space that space straight f left parenthesis straight x right parenthesis equals open vertical bar straight x minus 3 close vertical bar space has space no space derivative space at space straight x equals 3.
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    134. If space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x space sin 1 over straight x comma space space space straight x not equal to 0 end cell row cell 0 space space space space space space space space space comma space space space straight x equals 0 end cell end table close the space show space that space straight f space is space not space differentiable space at space straight x equals 0.
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    135. Syntax error from line 1 column 169 to line 1 column 176.
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    136. Examine the derivability of the following functions :
    |x| at x = 0
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    137. Examine the derivability of the following functions :
    |x2| at x = 0

    Let f(x) = |x|2
    straight L. straight H. straight D equals Lt with straight x rightwards arrow 0 to the power of minus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 0 right parenthesis over denominator straight x minus 0 end fraction equals Lt with straight x rightwards arrow 0 to the power of minus below fraction numerator open vertical bar straight x close vertical bar squared minus 0 over denominator straight x minus 0 end fraction space space space space space space space space space space space space space space space space space space left square bracket because space straight f left parenthesis 0 right parenthesis equals open vertical bar 0 close vertical bar squared equals 0 right square bracket space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 to the power of minus below open vertical bar straight x close vertical bar squared over straight x equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar 0 minus straight h close vertical bar squared over denominator 0 minus straight h end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket Put space straight x equals 0 minus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 0 to the power of minus right square bracket space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator straight h squared over denominator negative straight h end fraction equals Lt with straight x rightwards arrow 0 below left parenthesis negative straight h right parenthesis equals 0 straight R. straight H. straight D equals Lt with straight x rightwards arrow 0 to the power of plus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 0 right parenthesis over denominator straight x minus 0 end fraction equals Lt with straight x rightwards arrow 0 to the power of plus below open vertical bar straight x close vertical bar squared over straight x space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar 0 plus straight h close vertical bar squared over denominator 0 plus straight h end fraction space space left square bracket Put space straight x equals 0 plus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 0 to the power of plus right square bracket space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below straight h squared over straight h equals Lt with straight x rightwards arrow 0 below straight h over straight h equals 0 therefore space straight L. straight H. straight D not equal to straight R. straight H. straight D. equals 0 therefore space straight f left parenthesis straight x right parenthesis equals open vertical bar straight x close vertical bar squared space is space derivalbe space at space straight x equals 0 space and space has space the space derivative space 0.
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    138. A function f is defined as straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 1 plus straight x comma space if space straight x less or equal than 2 end cell row cell 5 minus straight x comma space if space straight x greater than 2 end cell end table closeShow that f is not differentiable at x = 2.
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    139. Examine the derivability of the following function:
    straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x squared space sin 1 over straight x comma space straight x not equal to 0 end cell row cell space space space space space space space 0 space space space comma space space space space straight x equals 0 end cell end table close
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    140. Examine the derivability of the following function:
    straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 3 minus 2 straight x comma space straight x less than 2 end cell row cell 3 straight x minus 7 comma space straight x greater or equal than 2 end cell end table close at space straight x equals 2
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