Prove that: 1. 2ab≤a2+b2 2. ab+ac+bc≤a2+b2+c2 If a, b and c are all integers.

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nicekikah

2021-08-20

Prove that:
1.

2 a b \leq a^{2} + b^{2}

a b + a c + b c \leq a^{2} + b^{2} + c^{2}

If a, b and c are all integers.

comentezq

Skilled2021-08-21Added 106 answers

1. a and b are integers, therefore a-b is an integer. An integers square is always greater then or equal to 0.

0 \leq {(a - b)}^{2}

{(a - b)}^{2} = a^{2} - 2 a b + b^{2}

0 \leq a^{2} - 2 a b + b^{2}

2 a b \leq a^{2} - 2 a b + b^{2} + 2 a b

2 a b \leq a^{2} + b^{2}

2. Since a, b and c are integers, any difference between them is also an integer. An integers square is always greater then or equal to 0.

0 \leq {(a - b)}^{2} + {(b - c)}^{2} + {(c - a)}^{2}

0 \leq a^{2} - 2 a b + b^{2} + b^{2} - 2 b c + c^{2} + c^{2} - 2 a c + a^{2}

2 a b + 2 b c + 2 a c \leq 2 a^{2} + 2 b^{2} + 2 c^{2}

a b + b c + a c \leq a^{2} + b^{2} + c^{2}

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