Since the second derivative is defined in terms of the first derivative, if f'' does not exist, does it imply that f' does not exist also? Similarly, if f' does not exist, does it imply that f'' not exist?
If the second derivative not exist, does the first derivative not exist also
calculusderivatives
Related Solutions
Really $|x|$ is just a convenient shorthand for what's really a piecewise defined function. If you allow piecewise defined functions, then it's rather easy to come up with examples.
$$f(x)=x^2,x>0$$ $$f(x)=-x^2,x<0$$
does the trick quite nicely at $x=0$. To make a function that has no second derivative at finitely/countably(?) many arbitrary points you can simply shift and add multiple functions like these.
The Weierstrass function is a famous example of a function which is everywhere continuous, but nowhere differentiable.
Let us write $f(x)$ for this function. Then the function given by
$$ F(x)=\int_0^xf(t)dt $$
is differentiable everywhere but twice differentiable nowhere.
See this answer for graphs of both functions.
Best Answer
Think about $$ f(x) = x|x| \ . $$