A Root Gap Bound

★ 8/10calculusConvexityMonotonicity

Problem

The function f(x)=(x+b)(ex−a)f(x)=(x+b)\left(\mathrm e^x-a\right) (b>0b>0) has tangent line

(e−1)x+ey+e−1=0(\mathrm e-1)x+\mathrm e y+\mathrm e-1=0

at the point (−1,f(−1))(-1,f(-1)).

1. Find aa and bb. 2. Suppose the equation f(x)=mf(x)=m has two real roots x1<x2x_1<x_2. Prove that

x2−x1⩽1+m(1−2e)1−e.x_2-x_1\leqslant 1+\frac{m(1-2\mathrm e)}{1-\mathrm e}.

Answer

Solution

Difficulty8/10
Topicscalculus, Convexity, Monotonicity

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