continuous function

Question

Let f be the function defined by the formula f={-a+(a+b)^1/3, if x>0 , 3 if x=3, x^3+a(x-1) if x<3}

Find the sum of parameters a and b for which f is continuous.

Answer 1

For the function to be continuous, all three portions of it must connect together at the same point. The point in question (based on the way the function is stated) is at x = 3. Since the middle expression specifies that at x = 3, the function value is 3, then that point (3, 3) is the point that all three pieces need to go through.

So, set the first and third parts equal to the function value of 3. The third part also has an x-value ion it, let that = 3 (the x-value is 3 as well as the y value):

-a+(a+b)^1/3 = 3

and

3^3+a(3-1) = 3 <- this can be solved for a: 27 + 2a = 3, 2a = -24, a = -12 then plug that into the first part:

-(-12) + ((-12) + b )^1/3 = 3, 12 + (b - 12)^1/3 = 3, (b - 12)^1/3 = -9, (b - 12) = (-9)^3, b - 12 = -729, b = -717