Homework Help: Finding constants in a piecewise function that allow the function to be differentiabl

1. Oct 9, 2008

souldoutt

1. The problem statement, all variables and given/known data
Find the values of the constants a and b such that the function f(x) is differentiable on R

2. Relevant equations
f(x) = ax2 if x < 2

f(x) = -4(x-3) + b if x >= 2

3. The attempt at a solution
ax2 = -4(x-3) + b
2xa = -4x
a = -2

I believe that I need to equate the equations but with a value of a how do I find b and then prove that it's differentiable?

2. Oct 9, 2008

tiny-tim

Welcome to PF!

Hi souldoutt! Welcome to PF!

(however did you get 2xa = -4x? )

Hint: the only problem is at x = 2.

So just bung x = 2 in, and check for continuity and differentiability.

3. Oct 11, 2008

souldoutt

Re: Finding constants in a piecewise function that allow the function to be different

I got the 2xa = -4x by differentiating both sides.
But with 2 separate unknown constants, how would i solve for them? I can plug the value x = 2 into the equations but i wont get an answer to confirm whether the second part of the function is actually starting from x = 2.

Then wouldn't I still need to have the constants in order to check whether the slopes of the tangents are the same? (therefore differentiable)?

thanks for the welcome too.

4. Oct 12, 2008

tiny-tim

Hi souldoutt!

For continuity, if you put x = 2, you get f(2) = 4a and = 4 + b,

so your continuity equation is 4a = 4 + b.

And similarly for differentiability you get f'(2) = 4a = -8.

So … ?

5. Oct 12, 2008

HallsofIvy

Re: Finding constants in a piecewise function that allow the function to be different

Yes, a must be -2. Now, you put a= -2 and x= 2 in the first equation you have -4= 4+ b. Solve for b.

To prove it is differentiable, with the correct values for a and b, Look at the difference quotient limit.

6. Oct 13, 2008

souldoutt

Re: Finding constants in a piecewise function that allow the function to be different

It should be -8 instead of -4 right? Because it is ax2 which = -8 when the numbers are plugged in.

Then, once I have the values of a and b the difference quotient limit to check for differentiability is the difference quotient limit of the derivatives correct?