Need help finding a b and c so that piecewise function f is differential everywhere

[prodigiii]

f(x) = -12x - 5 if x < -2
f(x) = ax^2 + bx + c if -2 <= x <= 2
f(x) = 4x - 4 if x > 2


Find a, b, and c so that f is differentiable everywhere.


(Hint: Find a system of four equations of three unknowns. This system will still have a solution.)


I've been working on this problem for 3 days and I can't find a solid answer. Its due tomorrow and I'm freaking out. Can anyone point me in the right direction?

 

[Deleted member 4993]

f(x) = -12x - 5 if x < -2
f(x) = ax^2 + bx + c if -2 <= x <= 2
f(x) = 4x - 4 if x > 2


Find a, b, and c so that f is differentiable everywhere.


(Hint: Find a system of four equations of three unknowns. This system will still have a solution.)


I've been working on this problem for 3 days and I can't find a solid answer. Its due tomorrow and I'm freaking out. Can anyone point me in the right direction?

What are the conditions of "differentiability" of a function at a given point?

 

[HallsofIvy]

A function is continuous at a given point if and only if the limits "from above" and "from below" are both equal to the value of the function at that point. Even if a function is differentiable at a point, it is not necessarily true that the derivative itself is continuous. However, it is still true that if the derivative of the limit "from above" and "from below" of the derivative exist they must be equal.

What are the only two points where it is not obvious that this function is differentiable for all a, b, c? What are the limits "from above" and "from below" of the function and its derivative at each of those points? Those will be your "four equations" in the "three unknowns", a, b, and c.

 

[stapel]

Given the following function:

. . . . .\(\displaystyle f(x)\, =\, \begin{cases}-12x\, -\, 5&\mbox{if }\, x\, <\, -2\\ax^2\, +\, bx\, +\, c&\mbox{if }-2\, \leq\, x\, \leq\, 2\\4x\, -\, 4&\mbox{if }x\, > 2\end{cases}\)

Find values of a, b, and c such that f will be differentiable everywhere. (Hint: Find a system of four equations of three unknowns. This system will still have a solution.)

I've been working on this problem for 3 days... Can anyone point me in the right direction?

The "right direction" is contained in the "Hint".

You've worked with piecewise functions back in algebra. You've worked with graphing linear and quadratic functions back in algebra. You know that, to be differentiable, the pieces of the function have to be continuous. They also have to meet up with the same slopes. You know how to evaluate the first and third "halves" of the function at their endpoints to get numerical values. You know how to evaluate the middle "half" at each of its endpoints to get algebraic expressions. Since you know the ends of each "half" must meet, you have set up the two algebraic equations created by this requirement.

Now you're in calculus, so you know about derivatives. You know that the "halves" must meet up with the same slopes, so you differentiated each of the three "halves". You evaluated the first and third "halves" to find their slope values. You evaluated the middle "half" at each of its endpoints to find its two slope expressions. Then you applied the derivative requirement to create two more algebraic equations.

You have now been working with these four equations for three days, trying to solve the algebraic system of linear equations. Please reply showing all of your work so far, starting with the first algebra you did (outlined in the first paragraph above), so we can see what's going on and where you're getting stuck.

Please be complete. Thank you!

 

[prodigiii]

I appreciate the pointers. I did end up figuring it all out by using a system of equations to find B, using derivatives to find A, and then solving for what C is using what I have. Thank you guys for the help.