For what value of the constant c is the function f continuous on (−∞,∞) ? f(x)={cx2+4xif x&lt;5x3−cxif x≥5

Question

For what value of the constant c is the function f continuous on (−∞,∞) ?  f(x)={cx2+4xif x&amp;lt;5x3−cxif x≥5

Susan Yang · Accepted Answer

The function f is defined by: The definition of the function changes at x=5 So the function is said to be continuous if the left hand limit and the right hand limit of the function are equal and same as the value of the function at the point. f(x)={cx2+4xif x&amp;lt;5x3−cxif x≥5 The left hand limit of the function is given by: The right hand limit of the function is given by: And the value of the function is same as the right hand limit of the function. limx→5−f(x)=limx→5−(cx2+4x) limx→5−f(x)=c(5)2+4(5) limx→5−f(x)=25c+20 limx→5−f(x)=25c+20 limx→5+f(x)=limx→5+(x3−cx) limx→5+f(x)=125−5c Hence, for the function to be continuous the limits must be equal: Hence the function is continuous when c=3.5 25c+20=125−5c 25c+5c=125−20 30c=105 c=10530 c=3.5 Answer: The function f is continuous when for c=3.5

For what value of the constant c is the function

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