对一切实数x,若ax² bx c≥0恒成立,且a

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由题意 a>0且 Δ=b² - 4ac ≤ 0
∴ c ≥ b²/(4a)
∵a0∴(a+b+c)/(b-a)
≥ [ a+b+ b²/(4a) ] /(b - a)
【分子分母同除以 a】
= [ 1+ b/a + 1/4(b/a)² ] (b/a - 1)
令 x = b/a >1= (1+x + x²/4) / (x-1)
= [ 1/4 (x-1)² + 3/2 (x-1) + 9/4 ] / (x-1)
= 1/4 (x-1) + (9/4)/(x-1) + 3/2
≥ 2 √(1/4*9/4) + 3/2
= 3当且仅当 1/4(x-1) = 9/4 (x-1) 即 x = 4 时取等号
故最小值为 3