Check if the following functions are differentiable

[Pole]

Homework Statement

Check if the following functions f : ℝ → ℝ are differentiable:
$\displaystyle f(x)=|(x-1)^{2}(x+1)^{3}|$

$\displaystyle f(x)=|x^{2}-\pi^{2}|sin^{2}x$

Homework Equations

The Attempt at a Solution

I don't know what the condition should be, I've searched a lot of topics concerning this problem but they all contain reference to some points or intervals at which it should be checked.
That's why I have no idea how to start and what definition has to be used.

Thanks in advance!

 

[Simon Bridge]

Do you know what a derivative is?
What it means for the derivative to exist at a point?

Notice that these are absolute value functions.
What does that do to the values?
Have you tried plotting any of them?

 

[Harrisonized]

They are "obviously" differentiable, except at a set of isolated points. Check those points. Show that the limit at x (from the definition of the derivative) does not exist. You can show this by noting that if a limit exists, it is unique.

 

[Pole]

They are "obviously" differentiable, except at a set of isolated points. Check those points. Show that the limit at x (from the definition of the derivative) does not exist. You can show this by noting that if a limit exists, it is unique.

After hours of reading I finally got to that, but the problem now is how to divide $\displaystyle f(x)=|(x-1)^{2}(x+1)^{3}|$ into cases so that it'd be so easily calculated as eg. $|x-2|$ where we have 2 cases - for x > 2 and x < 2, then make the derivative of both functions and check whether it's the same or not.

 

[Harrisonized]

If you square something, it's always positive, so you can remove the absolute value sign.

 

[SammyS]

Homework Statement

Check if the following functions f : ℝ → ℝ are differentiable:
$\displaystyle f(x)=|(x-1)^{2}(x+1)^{3}|$

$\displaystyle f(x)=|x^{2}-\pi^{2}|\sin^{2}x$

Homework Equations

The Attempt at a Solution

I don't know what the condition should be, I've searched a lot of topics concerning this problem but they all contain reference to some points or intervals at which it should be checked.
That's why I have no idea how to start and what definition has to be used.

Thanks in advance!

Hello Pole . Welcome to PF !

Have you learned about the differentiability of any broad classes of functions, such as polynomials, sinusoidal (sin(x) & cos(x)) functions, exponential functions, etc. ?

The absolute value function, |x|, has a piecewise definition, so when it is combined with other functions, you should check at those places at which the function is 'pieced together' .

 

[SammyS]

If you square something, it's always positive, so you can remove the absolute value sign.

Almost right. Actually, if you square something, it's always non-negative (it can be zero), so you can remove the absolute value sign.

So, the first function can be written as:

$f(x)=(x-1)^2\left|(x+1)^3\right|$​

The piecewise definition of the absolute value function is:

$\displaystyle |x|=\left\{\matrix{x\,,\text{ if }\ x\ge0\\ -x\,,\text{ if }\ x<0} \right.$​