Find so that the following function is continuous on any interval: f(x)=kx 0<x<4, and f(x) = 5x^2 if 4<x
What is f(4)?
at f(4), one side = 4k, the other 80
to make them continous, k must be 20
I agree with bobpursley if 0<x<=4 or 4<=x
To find the value of k such that the function f(x) = kx is continuous on any interval, we need to ensure that the limit of f(x) as x approaches 4 from both sides (x → 4-) and (x → 4+) is equal.
First, we find the limit of f(x) as x approaches 4 from the left side (x → 4-):
lim(x → 4-) kx = 4k
Next, we find the limit of f(x) as x approaches 4 from the right side (x → 4+):
lim(x → 4+) 5x^2 = 5*4^2 = 5*16 = 80
For the function to be continuous, the limits from both sides must be equal:
4k = 80
To find the value of k, we solve the equation for k:
k = 80/4
k = 20
Therefore, for the function f(x) = kx to be continuous on any interval, the value of k must be 20.