Question

Examine the derivability of the following function:
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x minus 1 comma space space space straight x less than 2 end cell row cell 2 straight x minus 3 comma space straight x greater or equal than 2 end cell end table close at space straight x equals 2$

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Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x minus 1 comma space space space straight x less than 2 end cell row cell 2 straight x minus 3 comma space straight x greater or equal than 2 end cell end table close straight L. straight H. straight D equals Lt with straight x rightwards arrow 2 to the power of minus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 2 right parenthesis over denominator straight x minus 2 end fraction equals Lt with straight x rightwards arrow 2 to the power of minus below space fraction numerator left parenthesis straight x minus 1 right parenthesis minus left parenthesis 4 minus 3 right parenthesis over denominator straight x minus 2 end fraction equals Lt with straight x rightwards arrow 2 to the power of minus below fraction numerator straight x minus 2 over denominator straight x minus 2 end fraction equals Lt with straight x rightwards arrow 2 to the power of minus below left parenthesis 1 right parenthesis equals 1 straight R. straight H. straight D equals Lt with straight x rightwards arrow 2 plus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 2 right parenthesis over denominator straight x minus 2 end fraction equals Lt with straight x rightwards arrow 2 to the power of plus below space fraction numerator left parenthesis 2 straight x minus 3 right parenthesis minus left parenthesis 4 minus 3 right parenthesis over denominator straight x minus 2 end fraction equals Lt with straight x rightwards arrow 2 to the power of plus below fraction numerator 2 straight x minus 4 over denominator straight x minus 2 end fraction space space space space space space space space space space space stack equals Lt with straight x rightwards arrow 2 to the power of plus below fraction numerator 2 left parenthesis straight x minus 2 right parenthesis over denominator straight x minus 2 end fraction equals 2 Lt with straight x rightwards arrow 2 to the power of plus below left parenthesis 1 right parenthesis equals 2 cross times 1 equals 2 therefore space 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 space is space not space dervable space at space straight x equals 2.

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