For what value of the constant c is the function f continuous on the interval negative infinity to positive infinity?
f(x)={cx2+7x if x<2
x3−cx if x> or =2
look at the "transition point"
in f(x) = cx^2 + 7x
f(2) = 4c + 14
in f(x) = x^3 - cx
f(2) = 8 - 2c
to be continuous the y values must be the same
4c + 14 = 8 - 2c
6c = -6
c = -1
To determine the value of the constant c for which the function f(x) is continuous on the interval from negative infinity to positive infinity, we need to find the values of c that make the two parts of the function match at the point x = 2.
First, let's evaluate the left-hand side of the function at x = 2:
f(2) = c(2^2) + 7(2) = 4c + 14
Now, let's evaluate the right-hand side of the function at x = 2:
f(2) = (2^3) - c(2) = 8 - 2c
For the function to be continuous at x = 2, the left-hand side and the right-hand side should be equal. Therefore, we set the expressions equal to each other and solve for c:
4c + 14 = 8 - 2c
Combining like terms, we get:
6c = -6
Dividing both sides by 6, we find:
c = -1
So, the constant c = -1 makes the function f(x) continuous on the interval from negative infinity to positive infinity.
To find the value of the constant c for which the function f is continuous on the interval from negative infinity to positive infinity, we need to ensure that the two parts of the function, namely cx^2 + 7x and x^3 - cx, are equal at x = 2.
First, let's find the value of f(x) as x approaches 2 from the left side (x < 2).
f(x) = cx^2 + 7x
f(2-) = c(2)^2 + 7(2) = 4c + 14
Now, let's find the value of f(x) as x approaches 2 from the right side (x ≥ 2).
f(x) = x^3 - cx
f(2+) = (2)^3 - c(2) = 8 - 2c
For the function f to be continuous at x = 2, the left and right limits should be equal.
So, f(2-) = f(2+)
4c + 14 = 8 - 2c
Now, let's solve this equation for c.
Combine like terms:
6c + 14 = 8
Subtract 14 from both sides:
6c = -6
Divide both sides by 6:
c = -1
Therefore, the constant c = -1 makes the function f continuous on the interval from negative infinity to positive infinity.