To determine the value of the constant c that makes the function f continuous, we need to analyze the function at the points where the definition changes. The function is defined as:
- f(x) = cx2 + 2x + 1 for x < 2
- f(x) = cx for x > 2
The function will be continuous at a point if the limit of f(x) as x approaches that point from both sides equals the function’s value at that point.
Step 1: Ensure Continuity at x = 2
We begin with ensuring the function is continuous at x = 2. We need to find:
-
Limit from the left:
lim (x -> 2-) f(x) = c(2)2 + 2(2) + 1 = 4c + 4 + 1 = 4c + 5 -
Limit from the right:
lim (x -> 2+) f(x) = c(2) = 2c
Setting these equal for continuity:
4c + 5 = 2c
Simplifying gives:
4c - 2c + 5 = 0
2c + 5 = 0
2c = -5
c = - rac{5}{2}
Step 2: Ensure Continuity at x = 3
Next, we check the continuity at x = 3:
We need to find:
-
Limit from the left:
lim (x -> 3-) f(x) = c(32) + 2(3) + 1 = 9c + 6 + 1 = 9c + 7 -
Limit from the right:
lim (x -> 3+) f(x) = c(3) = 3c
Setting these equal for continuity:
9c + 7 = 3c
Simplifying gives:
9c - 3c + 7 = 0
6c + 7 = 0
6c = -7
c = - rac{7}{6}
Conclusion
At this point, we derive two different values for c based on the respective continuity requirements:
– For the function to be continuous at x = 2, we find c = - rac{5}{2} and
– For x = 3, c = - rac{7}{6}.
Since these values are not equal, the function cannot be continuous at both points simultaneously for any single c. Therefore, there isn’t a single value for c that would make the function f continuous at both points x = 2 and x = 3.