For what value of the constant c is the function f continuous on the interval (-∞,∞).

solution:

f (x) has discontinuity at x=3

if it is continuous at x=3

then the continuous on (-/infty,/infty)

we have,

f (x) = / left / {cx^{2}+8; if x< 3, x^{3}-cx; if x/geq3

/lim_{x/rightarrow 3^-}f (x) = / lim_{x/rightarrow3^+}f (x)

/lim_{x/rightarrow3^-} (x^{2}+8x) = / lim_{x/rightarrow3^+} (x^{3}-cx)

(9c+24) = (227-3c)

9c+24=27-3c

9c+3c=27-24

12c=3

c=/frac{1}{4}