Question 7: Know that the function \(y = {\left( {x + a} \right)^3} + {\left( {x + b} \right)^3} – {x^3}\) there are two extreme points. Which of the following statements is true?
We have \(y = {x^3} + 3\left( {a + b} \right){x^2} + 3\left( {{a^2} + {b^2}} \right) x + {a^3} + {b^3}\)
\(y’ = 3{x^2} + 6\left( {a + b} \right)x + 3\left( {{a^2} + {b^2}} \right)\)
The function has two extremes if and only if y’ has two distinct solutions \( \Leftrightarrow \Delta ‘ = 18ab > 0 \Leftrightarrow ab > 0\)
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