Two Classics Combined

★ 7/10calculus Monotonicity Estimation inequality Casework

Problem

Let $f(x) = \mathrm{e}^x$ and $g(x) = \ln x$ .

1. If $h(x) = f(x) + a\,g(x)$ has a local minimum, find the range of possible values of the real number $a$ . 2. If $m > 0$ and

m^2 x^2 f(x-1) - (x+1)g(x) - mx \geqslant 0

holds for all $x > 0$ , find the range of possible values of $m$ .

Answer

Solution

Difficulty	7/10
Topics	calculus, Monotonicity, Estimation, inequality, Casework

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