MATH HOMEWORK Question

Question

for what value of the constant c is the function f continuous on (−∞ ∞)

f(x)={ cx^(2)+2x if x<5

{ x^(3)-cx if x>=5

c =

Answer 1

Continuity at x=5 means that the behavior of the function as it approaches 5 from the left is the same as the behavior as it approaches 5 from the right, and is also equal to the value at 5. There aren't any asymptotes or holes to deal with, so all it really means is "what would make these two parts of the function meet?"

Or, put another way, this question is really asking, "For which value of c does

cx^(2)+2x = x^(3)-cx if x=5

So, let's plug 5 in.

c(5)² + 2(5) = (5)³ -c(5)

25c+10 = 125-5c

And now it's a linear equation. Get the variables to one side and the constants to the other

30c = 115

divide by 30, and give your answer in whichever format your teacher prefers.

For this question, I'd most likely prefer the improper fraction 23 / 6.

But 3.83 with a bar over the final 3 to show it repeats is also accurate, as is the mixed number 3 and 5/6.