Give exact integer value.
Given the function f(x)
f(x) = {x^2 -x+1 if x ≤ 1
{kx^2 +1 if x >1
Find the value of k which makes the function continuous.
Give exact integer value.?
2016-03-25 3:25 pm
回答 (3)
2016-03-25 3:55 pm
If x ≤ 1, and x → 1, then f (x) = x^2 -x+1, → 1.
If x > 1 and x → 1, then f (x) = kx^2 + 1, → k + 1.
For continuity, these must be equal, requiring k = 0.
If x > 1 and x → 1, then f (x) = kx^2 + 1, → k + 1.
For continuity, these must be equal, requiring k = 0.
2016-03-25 3:30 pm
Need x^2 - x + 1 = kx^2 + 1 at x=1, so plug in and get 1 = k + 1 so k=0.
2016-03-25 4:03 pm
Examine the limit from the left of 1
lim x-->1- f(x) = lim x-->1- (x^2-x+1) = 1
Examine the limit from the right of 1
lim x-->1+ f(x) = lim x-->1+ (kx^2+1) = k+1
In order for the limit to exist at x=1 , we must have lim x-->1- f(x) = lim x-->1+ f(x)
k+1 = 1
k = 0
f(1) = 1
k = 0 makes f(x) continuous
lim x-->1- f(x) = lim x-->1- (x^2-x+1) = 1
Examine the limit from the right of 1
lim x-->1+ f(x) = lim x-->1+ (kx^2+1) = k+1
In order for the limit to exist at x=1 , we must have lim x-->1- f(x) = lim x-->1+ f(x)
k+1 = 1
k = 0
f(1) = 1
k = 0 makes f(x) continuous
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