let f(x)=x^(16/3) 019x^(33/7). Determine the largest n for which f is a member of the set C^n (n-times continuously differentiable)

Question

This is a problem from calculus 1 and I'm not sure what the process is here? Are we to attempt to take a derivative, use Rolle's Theorem or something else? I'm looking for a procedure more than an answer, but an answer would be helpful as well.

Answer 1

The given function has non-integer powers of x, which means we can differentiate by using power rule. The first-order derivative will have a term with a power of x exactly 1 less than the given function. The 2nd-order derivative will have a term with a power of x exactly 2 less than the original, etc.

We combine that idea with the fact that if a function contains a negative power of x (integer or non-integer) it will NOT be continuous at x = 0.

So for example, the cube root function, f(x) = x^1/3 is itself continuous but it is not continuously differentiable (i.e. it is continuously differentiable 0 times) because its derivative, f^'(x) = 1/3x^-2/3 is not continuous at x = 0.

g(x) = x^5/4 is continuously differentiable exactly once:

g^'(x) = 5/4x^1/4

Applying these concepts should allow you to determine a process for answering the question above.