A function f is defined as Show that f is not differentiable at x = 2.

Question

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.

Solution

Here space 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 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 fraction numerator left parenthesis 1 plus straight x right parenthesis minus 3 over denominator straight x minus 2 end fraction space space space space space space space space space space space left square bracket because space straight f left parenthesis 2 right parenthesis equals 1 plus 2 equals 3 right square bracket 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 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 1 straight R. straight H. straight D equals Lt with straight x rightwards arrow 2 to the power of 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 fraction numerator left parenthesis 5 minus straight x right parenthesis minus 3 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 minus straight x over denominator straight x minus 2 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 equals Lt with straight x rightwards arrow 2 to the power of plus below fraction numerator negative left parenthesis straight x plus 2 right parenthesis over denominator straight x minus 2 end fraction equals negative 1 therefore space space straight L. straight H. straight D not equal to straight R. straight H. straight D therefore space straight f space is space not space derivable space at space straight x equals 2.

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