p143_26

p143_26 - Problem 26 Page 143 I suppose you finished part...

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Unformatted text preview: Problem 26 Page 143 I suppose you finished part (a) and part (b). Then from part (b), you should get ym ym = 4(6 + β )3 27(4 + β )2 (1) For part (c), we need to determine the smallest value of β for which ym ≥ 4, hence √ 4(6 + β )3 ≥ 4 ⇐⇒ β ≥ 6 + 6 3 27(4 + β )2 √ Hence the smallest value for β is 6 + 6 3. ym ≥ 4 ⇐⇒ (2) Here is the details: First dividing 4 on both sides and moving 27(4 + β )2 on the right hand side: 4(6 + β )3 ≥ 4 ⇐⇒ (6 + β )3 ≥ 27(4 + β )2 2 27(4 + β ) (3) Then we may do a sub u = 4 + β , then we get (6 + β )3 ≥ 27(4 + β )2 ⇐⇒ (u + 2)3 ≥ 27u2 (4) then expand (u + 2)3 = u3 + 6u2 + 12u + 8, then we get (u + 2)3 ≥ u2 ⇐⇒ u3 + 6u2 + 12u + 8 ≥ 27u2 (5) Then move 27u2 on the left hand side u3 + 3u2 + 12u + 8 ≥ 27u2 ⇐⇒ u3 − 21u2 + 12u + 8 ≥ 0 huyu@math.byu.edu (6) ...
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