SOLUTION: Determine the values of a and b such that the following function is differentiable at 0 f(x) = ax3cos(1/x)+bx +b , if x < 0 sqrt(a+bx), if x>= 0 From what I know, I n

Question 1164224: Determine the values of a and b such that the following function is differentiable at 0
f(x) = ax3cos(1/x)+bx +b , if x < 0
sqrt(a+bx), if x>= 0
From what I know, I need to prove that the function is continuous at x=0 thus limf(x) as x approaches 0 must exist and must be equals to f(0) which led me to the equation b = sqrt(a) limf(x) as x approaches 0 from left and from right. But from there I do not know how to proceed. Please advise. Thank you very much!
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

In order to be differentiable at a point, the function must be both continuous at that point and the derivative must exist at that point.

In order to be continuous at a point, the limit from the left must equal the limit from the right.

So

must be equal to

So the first condition is

But the derivative must exist at zero as well, so using the definition of the derivative and the function value at zero from either side as determined by evaluation of the above limits, we have: