Finding Values of a & b for Differentiability of f(x)

[souldoutt]

Homework Statement

Find the values of the constants a and b such that the function f(x) is differentiable on R

Homework Equations

f(x) = ax² if x < 2

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

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?

 

[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.

 

[souldoutt]

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 won't 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.

 

[tiny-tim]

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 won't 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)?

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 … ?

 

[HallsofIvy]

Homework Statement

Find the values of the constants a and b such that the function f(x) is differentiable on R

Homework Equations

f(x) = ax² if x < 2

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

The Attempt at a Solution

ax² = -4(x-3) + b
2xa = -4x
a = -2

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.

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?

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

 

[souldoutt]

in the first equation you have -4= 4+ b.

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?