Question

For what value of λ is the function defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight lambda open parentheses straight x squared minus 2 straight x close parentheses comma space if space straight x less or equal than 0 end cell row cell 4 straight x plus 1 comma space space space space space space space if space straight x greater than 0 end cell end table close$
continous at x=0? What about continuity at x=1?

Solution

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight lambda open parentheses straight x squared minus 2 straight x close parentheses comma space if space straight x less or equal than 0 end cell row cell 4 straight x plus 1 comma space space space space space space space if space straight x greater than 0 end cell end table close At space straight x equals 0 space space space space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of minus below straight lambda open parentheses straight x squared minus 2 straight x close parentheses equals straight lambda left parenthesis 0 minus 0 right parenthesis equals 0 space space space space Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below left parenthesis 4 straight x plus 1 right parenthesis equals 4 left parenthesis 0 right parenthesis plus 1 equals 0 plus 1 equals 1 therefore Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis not equal to Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis

At x=1

space space space space space space Lt with straight x rightwards arrow 1 below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 below left parenthesis 4 straight x plus 1 right parenthesis equals 4 plus 1 equals 5 Also space space space space straight f left parenthesis 1 right parenthesis equals 4 plus 1 equals 5 therefore space Lt with straight x rightwards arrow 1 below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 1 right parenthesis

∴ f is continuous at x = 1 whatever value of λ be.

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