A and B constant Question Differentiable

Question

What constants a and b is the function

f(x)= a + be^(x-1) if x<1

fx= x+cos2(x-1) if x> or equal to 1

differentiable at x=1

Answer 1

f(x) = a + be^x-1, if x < 1

 

        = x + cos(2(x - 1)), if x ≥ 1

 

For f to be differentiable when x = 1, the following two conditions must be met:

 

(1).  f must be continuous at x = 1.  In other words, the two                        "pieces" of the graph have to fit together when x = 1.

 

        So, plugging 1 into the two given expressions and setting them         equal to each other yields:  a + be⁰ = 1 + cos0.

 

                                                        a + b = 2

 

(2).  The "one-sided" derivatives must be the same when x = 1.

 

         The derivatives of each of the given expressions are be^x-1 and                       1 - 2sin(2(x-1)) respectively.  These must be equal to each other when x = 1. 

 

          So, be⁰ = 1 - 2sin0    Therefore, b = 1

 

We have, a + b = 2 and b = 1.  SOLUTION:  a = b = 1