求导证明不等式有图

Question

Answer 1

其实结果对任意实数都成立.
函数|x|·e^(-x²)是偶函数,不妨对x ≥ 0证明.
f(x) = x·e^(-x²),f'(x) = (1-2x²)e^(-x²).
可知当0 ≤ x ≤ 1/√2时f'(x) ≥ 0,f(x)单调递增,f(x) ≤ f(1/√2) = e^(-1/2)/√2.
在1/√2 ≤ x时f'(x) ≤ 0,f(x)单调递减,f(x) ≤ f(1/√2) = e^(-1/2)/√2.