Question

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.

Answer 1

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.

Answer 2

1