or what value of the constant c is the function f continuous on (−∞,∞) where
f(x)={cx+7cx2−7 if x∈(−∞,3] if x∈(3,∞)
any polynomial is continuous on any interval.
Looks like you are missing part of the function definition. As long as f(3) is the same for both parts, you're in.
So, look up the missing part of the definition and solve the equation for
f(3) = lim f->3+ f(x) on the right.
To find the value of the constant c for which the function f is continuous on the entire interval (-∞, ∞), we need to make sure that the left and right limits of the function at x = 3 match the value of the function at x = 3.
1. Left Limit at x = 3:
lim(x->3-) f(x) = lim(x->3-) (cx + 7cx^2 - 7)
Here, we substitute x = 3 into the first part of the function (cx) since x ∈ (-∞, 3].
lim(x->3-) cx = c * 3 = 3c
Next, we substitute x = 3 into the second part of the function (7cx^2 - 7).
lim(x->3-) 7cx^2 - 7 = 7c(3)^2 - 7 = 63c - 7
Therefore, the left limit at x = 3 is: lim(x->3-) f(x) = 3c + 63c - 7 = 66c - 7.
2. Right Limit at x = 3:
lim(x->3+) f(x) = lim(x->3+) (cx + 7cx^2 - 7)
Since x ∈ (3, ∞), we substitute x = 3 into the second part of the function (7cx^2 - 7) only.
lim(x->3+) 7cx^2 - 7 = 7c(3)^2 - 7 = 63c - 7
Therefore, the right limit at x = 3 is: lim(x->3+) f(x) = 63c - 7.
To make the function continuous at x = 3, the left and right limits should equal the function value at x = 3.
lim(x->3-) f(x) = lim(x->3+) f(x) = f(3)
66c - 7 = 63c - 7
66c = 63c
3c = 0
From the equation 3c = 0, we can see that c = 0.
So, the value of the constant c for which the function f is continuous on (-∞, ∞) is c = 0.