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 x2 x2
To determine the value of the constant c that makes the function f(x) continuous on the interval negative infinity to positive infinity, we need to verify that the function does not have any discontinuities within this interval.
First, let's check for any potential points of discontinuity. Points of discontinuity occur where the function has a jump, a hole, or an asymptote (vertical or slant).
In this case, we have two pieces or "branches" for the function f(x):
For x < 2: f(x) = cx^2 + 7x
For x >= 2: f(x) = x^3 - cx
To determine if the function has any points of discontinuity, we need to analyze the behavior of each piece as x approaches 2 from the left (x < 2) and from the right (x >= 2).
1. x < 2 (left of 2):
For the function to be continuous at x = 2, the left piece, cx^2 + 7x, must approach the same value as the right piece, x^3 - cx, as x approaches 2 from the left. Let's evaluate these expressions:
lim(x -> 2-) cx^2 + 7x = c(2)^2 + 7(2) = 4c + 14
2. x >= 2 (right of 2):
For continuity at x = 2, the right piece, x^3 - cx, must approach the same value as the left piece, cx^2 + 7x, as x approaches 2 from the right. Evaluating these expressions:
lim(x -> 2+) x^3 - cx = 2^3 - c(2) = 8 - 2c
For the function to be continuous at x = 2, the left and right limits must be equal. Therefore, we equate the expressions and solve for c:
4c + 14 = 8 - 2c
Combine like terms:
6c + 14 = 8
Isolate the variable:
6c = 8 - 14
6c = -6
Solve for c:
c = -6/6
c = -1
Therefore, the constant c = -1 makes the function f(x) continuous on the interval negative infinity to positive infinity.