Question

Find a function that is 5 times differentiable but not 6 times differentiable at x=0

Answer 1

Answer:

The function $f(x) = x^{5+\frac{1}{3}}$ is 5 times differentiable but not 6 times differentiable at $x=0$.

Step-by-step explanation:

Consider the function $f(x) = x^{5+\frac{1}{3}}$, which is defined in all real numbers. Note that $f(x)$ is 5 times differentiable and $f^{(5)}(x) = (5+\dfrac{1}{3}) (4+\dfrac{1}{3}) (3+\dfrac{1}{3}) (2+\dfrac{1}{3}) (1+\dfrac{1}{3})x^{\frac{1}{3}}$.

But we cannot differentiate it again because $x^{\frac{1}{3}}$ is not differentiable in 0.