A Taylor Cutoff

★ 9/10Extrema calculus Monotonicity Symmetry

Problem

Let

f(x) = \ln(1+x) - x + \frac{1}{2}x^2 - kx^3, \qquad 0 < k < \frac{1}{3}.

1. Prove that on $(0, +\infty)$ , $f(x)$ has a unique critical point and a unique zero. 2. Let $x_1$ and $x_2$ be the critical point and the zero of $f(x)$ on $(0, +\infty)$ respectively.

Define $g(t) = f(x_1 + t) - f(x_1 - t)$ . Prove that $g(t)$ is decreasing on $(0, x_1)$ .
Compare $2x_1$ with $x_2$ , and prove your conclusion.

Answer

Solution

Difficulty	9/10
Topics	Extrema, calculus, Monotonicity, Symmetry

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