Calculus (Continuity)
posted by Mishaka on .
If the following function is continuous, what is the value of a + b?
f(x) = {3x^2  2x +1, if x < 0
a cos(x) + b, if 0 </= x </= pi/3
4sin^2(x), if x > pi/3
A. 0
B. 1
C. 2
D. 3
E. 4
I know that since the function is continuous, it should be equal to 1 at 0 and 3 at pi/3 (To follow the other two pieces of the function). From here, I am having a great deal of difficulty figuring out what coordinates would make the function work in this way. Any help is appreciated.

To be continuous, the functions should have the same value at the transition values of x
First transition value: x = 0
at x = 0, 3x^2  2x + 1 = 1
and acosx + b = acos(0) + b = a+b
since they wanted the value of a+b and we know its value is 1
we are done: a+b = 1
check:
at x= π/3
acosπ/3 + b = a/2 + b
and 4sin^2 (π/3) = 4(√3/2)^2 = 4(3/4) = 3
then a/2 + b = 3
a+2b = 6
solving with a+b=1, subtract them
b = 5
then a+5 = 1
a = 4
so when x=0 , first function is 1
2nd function is 4cos0 + 5 = 4+5 = 1 , good
when x=π/3
2nd function is 4cos(π/3) + 5 = 4(1/2) + 5 = 3
3rd function is 4sin^2 (π/3) = 4(√3/2)^2 = 3 , good!
So the values of a=4 and b=5
or
a+b=1
make the functions continuous.