How would you test if a function is differentiable for all real values?
I understand that to be differentiable a function must be continuous. I am unsure how I would test this in every situation and not just at one point.
I was looking over some problems and had no idea how to test all values for a function.
Which of the following function(s) is/are differentiable for all real numbers $x$?
I. $f(x) = \sec(\pi x)\tan(\pi x)$
II. $f(x) = 2 + \sqrt[3]{x}$
III. $f(x) = x + |\sin x|$
IV. $f(x) = -3x3^{-2x}$
How would one go about doing this type of problem?
Thank you
First of all, I would check the domain of the functions. E.g. $f(x)=\sec (\pi x) \tan (\pi x)$ is not defined on all of $\mathbb{R}$, so it cannot be differentiable on $\mathbb{R}$. If the functions are defined on all of $\mathbb{R}$, I would try to differentiate and check where the first derivatives exist: as an example, the derivative of the function $g(x) = 2+x^{\frac{1}{3}}$ is $g'(x) = \frac{1}{3}x^{-\frac{2}{3}}$, so $g'$ is not defined at $x=0$. This means $g$ is not differentiable on $\mathbb{R}$. You may try these two steps with the remaining functions.