A lovely calculus proof problem! (National Taiwan University entrance exam)

Overview

blackpenredpen proves two interlacing-root results from a National Taiwan University entrance-exam problem. Rolle’s theorem establishes that a root of f' = -g lies between consecutive roots of f; for the second result, applying Rolle’s theorem to h(x) = xg(x), then using the product rule and the absence of intervening roots of g, establishes a root of f between consecutive roots of g.

Key takeaways

If f'(x) = -g(x), every pair of consecutive roots of f contains a root of g, by applying Rolle’s theorem directly to f.
For the condition (xg(x))' = xf(x), the useful function for Rolle’s theorem is h(x) = xg(x), since its endpoint values vanish at roots of g.
The second proof depends on consecutiveness: it guarantees g(c) ≠ 0 at the interior point supplied by Rolle’s theorem.
The product-rule identity cg'(c) + g(c) = 0, together with g(c) ≠ 0, rules out c = 0 and makes cf(c) = 0 sufficient to conclude f(c) = 0.

Chapters

0:00 First Interlacing Result: Apply Rolle’s Theorem to f

Let a < b be consecutive roots of f, so f(a) = f(b) = 0 and f has no roots between them.
Because f is differentiable, it is continuous on [a,b] and differentiable on (a,b), satisfying Rolle’s theorem’s hypotheses.
Rolle’s theorem gives c in (a,b) with f'(c) = 0; since f'(x) = -g(x), this means g(c) = 0.

4:35 Second Result: Apply Rolle’s Theorem to h(x) = xg(x)

For consecutive roots a < b of g, define h(x) = xg(x); then h(a) = h(b) = 0.
Applying Rolle’s theorem to h gives c in (a,b) such that h'(c) = 0.
The exam condition h'(x) = xf(x) therefore gives cf(c) = 0, but an additional step is needed to show c is nonzero.

8:10 Use g(c) ≠ 0 to Rule Out c = 0 and Finish

The product rule gives h'(c) = cg'(c) + g(c) = 0.
Because c lies between consecutive roots of g, g(c) ≠ 0; the product-rule equation therefore rules out c = 0.
With c ≠ 0, the condition cf(c) = 0 forces f(c) = 0, proving that f has a root between the consecutive roots of g.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, blackpenredpen.