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Continuity and Differentiability

Short 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. $Prove space that space straight f left parenthesis straight x right parenthesis equals open vertical bar straight x minus 3 close vertical bar space has space no space derivative space at space straight x equals 3.$

Here f(x) = |x - 3|

straight L. straight H. straight D equals Lt with straight x rightwards arrow 3 to the power of minus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 3 right parenthesis over denominator straight x minus 3 end fraction equals Lt with straight x rightwards arrow 3 to the power of minus below fraction numerator open vertical bar straight x minus 3 close vertical bar minus 0 over denominator straight x minus 3 end fraction space space space space space space left square bracket because space straight f left parenthesis 3 right parenthesis equals open vertical bar 3 minus 3 close vertical bar equals open vertical bar 0 close vertical bar equals 0 right square bracket space space space space space space space space space space space equals Lt with straight x rightwards arrow 3 to the power of minus below fraction numerator open vertical bar straight x minus 3 close vertical bar over denominator straight x minus 3 end fraction space space space space space space space space space space left square bracket Put space straight x equals 3 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 3 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 open vertical bar 3 minus straight h minus 3 close vertical bar over denominator 3 minus straight h minus 3 end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar negative straight h close vertical bar over denominator negative straight h end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator straight h over denominator negative straight h end fraction equals negative 1 straight R. straight H. straight D equals Lt with straight x rightwards arrow 3 to the power of plus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 3 right parenthesis over denominator straight x minus 3 end fraction equals Lt with straight x rightwards arrow 3 to the power of plus below fraction numerator open vertical bar straight x minus 3 close vertical bar minus 0 over denominator straight x minus 3 end fraction space space space space space space space space space space space equals Lt with straight x rightwards arrow 3 to the power of plus below fraction numerator open vertical bar straight x minus 3 close vertical bar over denominator straight x minus 3 end fraction space space space space space space space space space left square bracket Put space straight x equals 3 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 3 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 fraction numerator open vertical bar 3 plus straight h minus 3 close vertical bar over denominator 3 plus straight h minus 3 end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar straight h close vertical bar over denominator straight h end fraction equals Lt with straight h rightwards arrow 0 below straight h over straight h equals 1 therefore straight L. straight H. straight D not equal to straight R. straight H. straight D therefore space straight f left parenthesis straight x right parenthesis equals open vertical bar straight x minus 3 close vertical bar space is space not space derivable 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

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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 close

Show 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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