Let f(x) be the function
f(x)={x2−c for x<4,
{3x+4c for x≥4.
Find the value of c that makes the function continuous.
x^2-c and 3 x + 4 c
must be the same on both sides when x = 4
16 - c = 12 + 4 c
4 = 5 c
c = 4/5
To find the value of c that makes the function continuous, we need to ensure that the limits from the left and right side of x = 4 match.
First, let's find the limit of f(x) as x approaches 4 from the left side. For x < 4, we have f(x) = x^2 - c. So we need to find the limit of this expression as x approaches 4 from the left:
lim (x→4-) (x^2 - c)
Taking the limit, we plug in x = 4 and evaluate:
lim (x→4-) (16 - c) = 16 - c
Next, let's find the limit of f(x) as x approaches 4 from the right side. For x ≥ 4, we have f(x) = 3x + 4c. So we need to find the limit of this expression as x approaches 4 from the right:
lim (x→4+) (3x + 4c)
Taking the limit, we plug in x = 4 and evaluate:
lim (x→4+) (3(4) + 4c) = 12 + 4c
For the function to be continuous at x = 4, the limits from the left and right sides should match. So we set the left limit equal to the right limit:
16 - c = 12 + 4c
To find the value of c, we can solve this equation. Rearranging gives:
16 = 12 + 5c
Subtracting 12 from both sides:
4 = 5c
Dividing both sides by 5:
c = 4/5
Thus, the value of c that makes the function continuous is c = 4/5.