A = |x - 1| + |x + 5| + (x - 2)2 + 2017
A = |x - 1| + |x + 5| + |(x - 2)2| + 2017
A = |x - 1| + |x + 5| + |x2 + 4 - 4x| + 2017
Áp dụng bđt |a| + |b| + |c| \(\ge\)|a+b+c| ta có:
A = |x - 1| + |x + 5| + |x2 + 4 - 4x| + 2017 \(\ge\)|x - 1 + x + 5 + x2 + 4 - 4x| + 2017
A\(\ge\) |x2 - 2x + 8| + 2017
A \(\ge\) |x2 - x - x + 1 + 7| + 2017
A\(\ge\) |(x - 1)2 + 7| + 2017
A\(\ge\) (x - 1)2 + 2024
Dấu "=" xảy ra khi x - 1 \(\ge\)0; x + 5 \(\ge\)0
=> x \(\ge\)1; x \(\ge\)-5
=> x \(\ge\)1
Vậy GTNN của A là 2024 khi x = 1
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