Question 1184358


No, your function is not smooth in the domain [0, 25].  In fact, it's not even continuous.

What makes it continuous is the following definition: 


{{{f(x)=system(matrix(3,2, 2,"," 0<=x <=e, ln(x^2),"," e < x < e^3, x+6 - e^3, "," e^3 <=x <= 25))}}}.

But even with the correct definition it still is not smooth, as the function is not differentiable at {{{x = e}}} and {{{x = e^3}}} 

At {{{x = e}}}, the left-hand derivative is {{{"f'"[l] (e)=0}}}, while the right-hand derivative is {{{"f'"[r] (e)=2/e}}}, so the derivative at {{{x=e}}}does not exist.


At {{{x = e^3}}}, the left-hand derivative is {{{"f'"[l] (e^3)=2/e^3}}}, while the right-hand derivative is {{{"f'"[r] (e^3) = 0}}}, so the derivative at {{{x=e^3}}}does not exist.