Solving Problems Involving Differentiability of a Function

[Yankel]

Hello,

How do I solve this kind of problems ?

For which values of x the next function is "differentiable" ?

View attachment 530

I know it has something to do with the existent of the one sided limits, but which limits should I be calculating exactly ?

Thanks !

 

[MarkFL]

I would first check for continuity. Is the function continuous over the entire domain?

 

[Yankel]

well it ain't, there is a problem with x=1,2 and I think also x=0

 

[MarkFL]

Can you state why you think there are problems at the x-values you give?

 

[Deveno]

checking for continuity is a good idea, but in this particular problem it doesn't rule out any points for us, f is continuous everywhere:

$$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} 2 = 2$$
$$\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} \frac{2}{x^2} = 2$$

so f is continuous at 1.

$$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} \frac{2}{x^2} = \frac{2}{4} = \frac{1}{2}$$
$$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} \frac{1}{x} = \frac{1}{2}$$

so f is continuous at 2.

there's no problem at 0, f(0) = 2.

instead, you should look at places where the derivative might not exist.

 

[HallsofIvy]

Hello,

How do I solve this kind of problems ?

For which values of x the next function is "differentiable" ?

View attachment 530

I know it has something to do with the existent of the one sided limits, but which limits should I be calculating exactly ?

Thanks !

Since the question asks about "differentiability" I would think that the obvious first thing to do would be to look up the definition of "differentiable". Fortunately, for a function of a single variable that is simply "has a derivative there" (for functions of more than one variable it is more complicated).

So what is the definition of "the derivative of f(x) at x= a"?

(You are completely wrong when you say the function is not continuous. I think you need to review the basic definitions.)

 

[MarkFL]

As stated, the function has no discontinuities. So next, check to see if:

$\displaystyle \lim_{x\to1^-}f'(x)=\lim_{x\to1^+}f'(x)$

and

$\displaystyle \lim_{x\to2^-}f'(x)=\lim_{x\to2^+}f'(x)$

What do you find and what is your conclusion?