Real Roots of a Composed Equation

★ 8/10functions Extrema calculus Monotonicity Vieta's Formulas

Problem

Let $f(x)=-\ln x+ax^2+bx-a-2b$ have two extreme points $x_1,x_2$ , where $-\dfrac 12<a<0<b$ and $f(x_2)=x_2>x_1$ . Find the number of real roots of the equation

2a\bigl[f(x)\bigr]^2+bf(x)-1=0.

Answer

Solution

Difficulty	8/10
Topics	functions, Extrema, calculus, Monotonicity, Vieta's Formulas

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