Lebl-Basic-Analysis-I · Problem 4.4.4

The Derivative

Mathematics · nth-root-odd

Problem

Let n∈Nn \in \mathbb{N} be odd and n≥3n \geq 3. Prove that every xx has a unique nnth root. That is, there exists a number yy such that yn=xy^n = x. Prove that the function defined by g(x):=yg(x) := y is differentiable except at x=0x = 0 and compute the derivative. Prove that gg is not differentiable at x=0x = 0.
Intuitive explanation

The key insight is that odd-power functions are strictly monotonic everywhere, which guarantees both existence and uniqueness of roots. For differentiability, you'll use the inverse function theorem—since f(y)=ynf(y) = y^n has a nonzero derivative away from y=0y = 0, its inverse gg is automatically differentiable there. At the origin, the derivative of gg would require f′(0)≠0f'(0) \neq 0, but f′(0)=0f'(0) = 0, so the inverse function theorem fails and you must check the limit definition directly to confirm non-differentiability.

Intuitive explanation

The key insight is that odd-power functions are strictly monotonic everywhere, which guarantees both existence and uniqueness of roots. For differentiability, you'll use the inverse function theorem—since f(y)=ynf(y) = y^n has a nonzero derivative away from y=0y = 0, its inverse gg is automatically differentiable there. At the origin, the derivative of gg would require f′(0)≠0f'(0) \neq 0, but f′(0)=0f'(0) = 0, so the inverse function theorem fails and you must check the limit definition directly to confirm non-differentiability.

Browse the problem library