Câu hỏi

Cho a,b,c là các số thực dương. Chứng minh rằng:

R. Roboctvx102

Giáo viên

Xác nhận câu trả lời

Giải thích

Ta có: . . Đồng thời: . . Do đó: . Đặt . Vì a,b,c là các số thực dương nên t>2. ⇒ 2 ab + a c + b c = 2 t 2 − 4 . Xét hàm số: với t>2. Ta có: ( vì t>2). Do đó f ′ ( t ) = 0 ⇔ t = 4 . Bảng biến thiên: Dấu bằng xảy ra khi và chỉ khi

Ta có:
$square root of a b open parentheses a plus c close parentheses open parentheses b plus c close parentheses end root equals square root of open parentheses a b plus b c close parentheses open parentheses a b plus a c close parentheses end root less or equal than fraction numerator a b plus b c plus a b plus a c over denominator 2 end fraction$ .
$rightwards double arrow square root of a b open parentheses a plus c close parentheses open parentheses b plus c close parentheses end root less or equal than fraction numerator 2 a b plus b c plus a c over denominator 2 end fraction$ .
Đồng thời:
square root of 4 plus 4 a squared plus 4 b squared plus c squared end root equals square root of 4 plus open parentheses 2 a squared plus 2 b squared close parentheses plus 2 a squared plus c squared over 2 plus 2 b squared plus c squared over 2 end root greater or equal than square root of 4 plus 4 a b plus 2 a c plus 2 b c end root .
rightwards double arrow square root of 4 plus 4 a squared plus 4 b squared plus c squared end root greater or equal than square root of 4 plus 2 open parentheses 2 a b plus a c plus b c close parentheses end root .
Do đó:

$fraction numerator 9 over denominator square root of a b open parentheses a plus c close parentheses open parentheses b plus c close parentheses end root end fraction minus fraction numerator 32 over denominator square root of 4 plus 4 a squared plus 4 b squared plus c squared end root end fraction greater or equal than fraction numerator 18 over denominator 2 a b plus a c plus b c end fraction minus fraction numerator 32 over denominator square root of 4 plus 2 open parentheses 2 a b plus a c plus b c close parentheses end root end fraction$ .
Đặt t equals square root of 4 plus 2 open parentheses 2 a b plus a c plus b c close parentheses end root . Vì a,b,c là các số thực dương nên t>2.
$\Rightarrow 2 ab + a c + b c = \frac{t ^{2} - 4}{2}$ .
Xét hàm số: $f open parentheses t close parentheses equals fraction numerator 36 over denominator t squared minus 4 end fraction minus 32 over t$ với t>2.
$rightwards double arrow f to the power of apostrophe open parentheses t close parentheses equals fraction numerator negative 72 t over denominator open parentheses t squared minus 4 close parentheses squared end fraction plus 32 over t squared equals fraction numerator 32 t to the power of 4 minus 72 t cubed minus 256 t squared plus 512 over denominator open parentheses t squared minus 4 close parentheses squared. t squared end fraction equals fraction numerator open parentheses t minus 4 close parentheses open parentheses 32 t cubed plus 56 t squared minus 32 t minus 128 close parentheses over denominator open parentheses t squared minus 4 close parentheses squared. t squared end fraction$
Ta có: ( vì t>2).
Do đó $f^{'} (t) = 0 \Leftrightarrow t = 4$ .
Bảng biến thiên:

$rightwards double arrow f open parentheses t close parentheses greater or equal than negative 5 rightwards double arrow fraction numerator 9 over denominator square root of a b open parentheses a plus c close parentheses open parentheses b plus c close parentheses end root end fraction minus fraction numerator 32 over denominator square root of 4 plus 4 a squared plus 4 b squared plus c squared end root end fraction greater or equal than negative 5$

Dấu bằng xảy ra khi và chỉ khi