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For what value of the constant c is the function f continuous on (-infinity, infinity)?

f(x)= cx^2 + 2x   if x < 3 and
         x^3 - cx     if x ≥ 3

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Han L. | Experienced Mathematics Tutor including Calculus I and SAT MathExperienced Mathematics Tutor including ...
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So, we know that for any value of c, for x < 3, the function is continuous and for x >= 3, the function is continuous. Therefore, it is the case we have to check for the continuity of the function at x = 3. f(x) is continuous at x = a if the following conditions are satisfied. (i) f(a) exits. (ii) limx->a f(x) exists and (iii) limx->a f(x) = f(a).
We know that for any value of c, the first two conditions are satisfied. So, to find c, assume that the third condition is satisfied in order to make the function continuous. now, limx->3 f(x) = f(3). By solving this equation, you will get c = 7/4 = 1.75.  Here's the worked out.
limx->3 f(x) = 9c+6
f(3) = 27-3c
9c+6 = 27-3c
12c = 21
c = 21/12 = 7/4 = 1.75