Problem 70 (difficulty: 4/10)

Calculate the maximum value of the function \(\displaystyle x^2 \cdot (1-x)\) for \(\displaystyle x\in[0,1].\)

Solution:

By the AM-GM inequality,

\(\displaystyle \sqrt[3]{x\cdot x\cdot(2-2x)}{\leq}\frac{x+x+(2-2x)}3. \)

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