KindiakMath

Harder than It Seems

Problem 1. Let $f$ be differentiable on $\mathbb R$ and

$\displaystyle \lim_{x \to -\infty} f(x) = \lim_{x \to \infty} f(x)= 0.$

Prove that there exists $c \in \mathbb R$ such that $f'(c) = 0$ .

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Solution. The main idea is that $f(x) \approx 0$ for sufficiently large $|x|$ .

If $f = 0$ identically, then $f' = 0$ , and in particular, $f'(0) = 0$ . Now suppose $f \neq 0$ . Then there exists $x_0 \in \mathbb R$ such that $f(x_0) \neq 0$ . By applying subsequent arguments on $\pm f$ , we may assume without loss of generality that $f(x_0) > 0$ .

Define $\epsilon_0 := f(x_0)/2$ . Use the limits to find $N > |x_0| \geq 0$ such that

$|x| \geq N \quad \Rightarrow \quad f(x) \leq |f(x)| < \epsilon_0.$

In particular, $\max\{ f(-N), f(N)\} < \epsilon_0 < f(x_0)$ . Apply the intermediate value theorem on the intervals $[-N, x_0]$ and $[x_0, N]$ respectively to find $c_1 \in (-N, x_0)$ and $c_2 \in (x_0, N)$ such that

$f(c_1) = f(c_2) = \epsilon_0.$

Now apply Rolle’s theorem to obtain $c \in (c_1,c_2) \subseteq \mathbb R$ such that $f'(c) = 0$ , as required.

—Joel Kindiak, 25 Oct 24, 2153H

Calculus, Exercises

combinatorics, machine-learning, math, mathematics, physics

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