Question 5: Given the function (y = {x^4} – frac{2}{3}{x^3} – {x^2}.) Which of the following statements is true?

Question 5: Given the function \(y = {x^4} – \frac{2}{3}{x^3} – {x^2}.\) Which of the following statements is true?

A. The function has two minimum values \( – \frac{2}{3}\) and \( – \frac{5}{{48}}\).

B. The function has a minimum value of \( – \frac{2}{3}\) and a maximum value of \( – \frac{5}{{48}}\)

C. The function has a minimum value of 0

D. The function has only one minimum value.

\(y = {x^4} – \frac{2}{3}{x^3} – {x^2} \Rightarrow y’ = 4{x^3} – 2{x^2} – 2x\ )

\(y’ = 0 \Rightarrow \left[\begin{array}{l}x=0\\x=1\\x=–\frac{1}{2}\end{array}\right\)[\begin{array}{l}x=0\x=1\x=–\frac{1}{2}\end{array}\right\)

Variation table.

The function has two minimum values \( – \frac{2}{3}\) and \( – \frac{5}{{48}}\).

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