The equation x6−5x4+16x2−72x+9=0 has(A) exactly two distinct real roots(B) exactly three distinct real roots(C) exactly four distinct real roots(D) six distinct real roots

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crazyrocketrz5z · Accepted Answer

You have:f(x)=x6−5x4+16x2−72x+9f′(x)=6x5−20x3+32x−72f&amp;nbsp;″(x)=30x4−60x2+32If you notice thatf&amp;nbsp;″(x)=30(x4−2x2+1)+2=30(x2−1)2+2≥2&amp;gt;0you can see that f&#039;(x) is strictly increasing. Together with&amp;nbsp;limx→−∞f′(x)=−∞&amp;nbsp;and&amp;nbsp;limx→∞f&#039;(x)=∞&amp;nbsp;this implies that there is exactly one root&amp;nbsp;x0&amp;nbsp;of&amp;nbsp;f′(x)Thus&amp;nbsp;f(x)&amp;nbsp;is decreasing on&amp;nbsp;(−∞,x0)&amp;nbsp;and increasing on&amp;nbsp;(x0,∞)Since&amp;nbsp;limx→∞f(x)=∞&amp;nbsp;and&amp;nbsp;f(1)=1−5+16−72+9=−51, we see that&amp;nbsp;f(x)&amp;nbsp;has both positive and negative values.Thus&amp;nbsp;f(x)&amp;nbsp;must have two real roots, one of them in the interval&amp;nbsp;(−∞,x0)&amp;nbsp;and another in the interval&amp;nbsp;(x0,∞)

The equation \(\displaystyle{x}^{{6}}-{5}{x}^{{4}}+{16}{x}^{{2}}-{72}{x}+{9}={0}\) has (A) exactly two distinct

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