Find all values of x for which the function is differentiable given a piecewise function.

Question

Given the piecewise function

          f(x) = {(x+1)²,    x≤0

                     2x + 1,   0<x<3

                     (4-x)²,   x≥3

 find, all values of x for which the function is differentiable.

How do I solve the question above? My teacher explained it today, but I was utterly and completely lost. If you could provide a relatively understandable explanation and a step-by-step solution, it would be much appreciated.
Thank you in advance!!!

Answer 1

Each piece has a derivative, in your example.

You just have to see if the derivatives at the endpoints of adjacent intervals exist and are the same. This requires, of course, that the function is continuous.

 

At 0, f(x) = 1 and immediately to the right of that, f(x) has a right limit also equal to 1, so continuity is assured. And the respective derivatives are also the same (namely 2).

 

At 3, f(x) = 1 and just to the left of that, the f(x) values head toward the limit value 7, so f is not continuous at 3. 

 

My conclusion: differentiable at 0, not differentiable at 3.

Answer 2

We need only check for differentiability at x = 0 and 3.  The graph consists of pieces of lines and parabolas, which are differentiable functions.  

 

 For x = 3, lim_x→3^-   f(x) = lim_x→3^-  (2x+1) = 7

 

                lim_x→3⁺  f(x) = lim_x→3⁺  (4-x)² = 1

 

     Since the one sided limits are not equal, the function is not continuous at x=3, So, the function can't be differentiable either.

 

  For x = 0, the function is continuous there.  Check to see if the derivative exists:

 

       f'(0) = lim_x→0 [(f(x) - f(0))/(x-0)]

 

         Checking the one-sided limits:

 

           lim_x→0^- [(f(x) - f(0))/(x-0)] = lim_x→0^- [((x+1)² - 1)/(x-0)]

 

                                                    =lim_x→0^-  [(x² + 2x)/x] = 2

 

           lim_x→0⁺ ((f(x) - f(0))/(x-0)] = lim_x→0⁺ [(2x+1 - 1)/x] = 2

 

Since the one-sided limits are equal, the function is differentiable when x = 0.  

 

So, the function is differentiable for all x except for x = 3.