Three Solutions and a Double Extreme

Problem

Let

f(x) = \left(x^3 - ax\right)\ln\left(x^2 + 1 - a\right), \qquad x \in \mathbb{R}.

1. If the equation $f(x) = 0$ has exactly $3$ real solutions, find the range of possible values of $a$ . 2. Under the conditions of part 1, is there a real $a$ such that $f(x)$ has exactly two critical points $x_1, x_2$ in $(0, 1)$ with $x_2 = 2x_1$ ? If so, find $a$ ; if not, explain why.

Answer

Solution

Difficulty	9/10
Topics	Extrema, calculus, Monotonicity

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