Find a C such that f(x) is continuous on the entire real line f(x) = { x^2 when x is equal to or less than 4 and c/4 when x is less than 4
To find a value C such that f(x) is continuous on the entire real line, we need to ensure that the two pieces of the function, x^2 and c/4, agree at the point x=4.
In other words, we need to find a C such that f(4) is defined and f(4) is equal to both x^2 and c/4.
For x ≤ 4, f(x) = x^2. So, for x=4, f(4) = 4^2 = 16.
For x < 4, f(x) = c/4. So, for x=4, f(4) = c/4.
To make f(x) continuous at x=4, we need f(4) to equal both 16 and c/4. Therefore, we set these two equal to each other:
16 = c/4
To solve for c, we can multiply both sides of the equation by 4:
64 = c
Therefore, the value of C that makes f(x) continuous on the entire real line is 64.
To find a value of C such that f(x) is continuous on the entire real line, we need to make sure that the limit of f(x) from the left (as x approaches 4) is equal to the value of f(4).
First, let's calculate the limit from the left (as x approaches 4):
lim(x->4-) f(x) = lim(x->4-) c/4 = c/4
To ensure continuity, we must have this limit equal the value of f(4):
f(4) = 4^2 = 16
Therefore, we set c/4 equal to 16 and solve for c:
c/4 = 16
c = 4 * 16
c = 64
Thus, the value of C that makes f(x) continuous on the entire real line is C = 64.
To find the value of C that would make the function f(x) = x^2 when x is equal to or less than 4, and f(x) = c/4 when x is greater than 4 continuous on the entire real line, we need to ensure that the function is continuous at x = 4.
To be continuous at x = 4, we should ensure that the limit of f(x) as x approaches 4 from the left is equal to the value of f(x) at x = 4, and the limit of f(x) as x approaches 4 from the right is also equal to the value of f(x) at x = 4.
First, let's find the limit of f(x) as x approaches 4 from the left (x < 4):
lim (x→4-) f(x) = lim (x→4-) (c/4) = c/4
Now, let's find the limit of f(x) as x approaches 4 from the right (x > 4):
lim (x→4+) f(x) = lim (x→4+) (x^2) = 4^2 = 16
To make the function continuous, the value of f(x) at x = 4 should be equal to both the left and right limits:
f(4) = c/4 = 16
Solving for c, we can multiply both sides by 4:
c = 16 * 4
c = 64
Therefore, the only value of c that would make the function f(x) continuous on the entire real line is c = 64.