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(Where a question asks you to list values, separate the values by commas if there are more than one. If there are no values, enter *NONE*.)
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Suppose that
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\( f(x) = x^4 - $a x^3 \).
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$PAR
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(A) List all the critical values of \(f(x)\).
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Note: If there are no critical values, enter 'NONE'.
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$BR
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\{ans_rule(50)\}
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$PAR
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(B) Use interval notation to indicate where \(f(x)\) is increasing.
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$BR
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$BBOLD Note: $EBOLD Use 'INF' for \(\infty\), '-INF' for \(-\infty\),
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and use 'U' for the union symbol.
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$BR
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Increasing: \{ans_rule(50)\}
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$PAR
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(C) Use interval notation to indicate where \(f(x)\) is decreasing.
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$BR
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Decreasing: \{ans_rule(50)\}
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$PAR
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(D) List the \(x\) values of all local maxima of
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\(f(x)\). If there are no local maxima, enter 'NONE'.
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$BR
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\(x\) values of local maximums = \{ans_rule(30)\}
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$PAR
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(E) List the \(x\) values of all local minima of
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\(f(x)\). If there are no local minima, enter 'NONE'.
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$BR
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\(x\) values of local minimums = \{ans_rule(30)\}
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(F) Use interval notation to indicate where \( f(x) \) is concave up.
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$BR
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Concave up: \{ans_rule(50)\}
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$PAR
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(G) Use interval notation to indicate where \( f(x) \) is concave down.
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$BR
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Concave down: \{ans_rule(50)\}
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$PAR
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(H) List the \(x\) values of all the inflection points of
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\(f\). If there are no inflection points, enter 'NONE'.
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$BR
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\(x\) values of inflection points = \{ans_rule(30)\}
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$PAR
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[` f(x) = x^4 - [$a] x^3 `].
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(A) List all the critical values of [`f(x)`].
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[_]{$crit}{50}
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(B) Use interval notation to indicate where [`f(x)`] is increasing. [@ helpLink("interval") @]*
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Increasing: [_]{$incr}{50}
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(C) Use interval notation to indicate where [`f(x)`] is decreasing.
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Decreasing: [_]{$decr}{50}
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(D) List the [`x`] values of all local maxima of
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[`f(x)`].
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[`x`] values of local maximums = [_]{$maxima}{30}
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(E) List the [`x`] values of all local minima of
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[`f(x)`].
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[`x`] values of local minimums = [_]{$minima}{30}
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(F) Use interval notation to indicate where [` f(x) `] is concave up.
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Concave up: [_]{$up}{50}
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(G) Use interval notation to indicate where [` f(x) `] is concave down.
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Concave down: [_]{$down}{50}
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(H) List the [`x`] values of all the inflection points of
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[`f`].
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[`x`] values of inflection points = [_]{$inflec}{30}
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(I) Use all of the preceding information to sketch a
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graph of \(f\). When you're finished, enter a "1" in the box
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graph of [`f`]. When you're finished, enter a "1" in the box
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below.
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$BR \{ans_rule(10)\}
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END_TEXT
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[_]{Real(1)}{10}
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END_PGML
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# Used in solution
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Context("Fraction");
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$am3=3*$a;
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$am6=6*$a;
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$am3d4=Fraction($am3,4);
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$ad2=Fraction($a,2);
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BEGIN_PGML_SOLUTION
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[``\begin{aligned}
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f'(x)&=4x^3-[$am3]x^2=x^2(4x-[$am3])\\
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f''(x)&=12x^2 - [$am6]x = x(12x-[$am6])
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\end{aligned}``]
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so [`f'(x)=0`] when [`x=0`] or [``x=[$am3d4]``]. [`f'(x)`] is a polynomial so it is dedefined at every [`x`]. Thus there are two critical values, [`x=0`] and [``x=[$am3d4]``].
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[``f'(x)\geq 0``] when [``x\geq [$am3d4]``] and [``f'(x)\leq 0``] when [``x\leq [$am3d4]``]. It follows that [`f(x)`] is increasing on the interval [`([$am3d4],\infty)`] and [`f(x)`] is decreasing on the interval [`(-\infty,[$am3d4])`].
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Therefore [`f(x)`] has an absolute minimum at [`x=[$am3d4]`], and it has no local maxima.
[`f''(x)=0`] when [`x=0`] and when [`x=[$ad2]`]. [`f''(x)<0`] when [`0<x<[$ad2]`], and [`f''(x)>0`] when [`x<0`] or [`x>[$ad2]`]. Thus [`f(x)`] is concave up on [`(-\infty,0)\cup([$ad2],\infty)`], concave down on [`(0,[$ad2])`], and it has inflection points where the concavity changes at [`x=0`] and [`x=[$ad2]`].
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END_PGML_SOLUTION
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ENDDOCUMENT(); # This should be the last executable line in the problem.
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ENDDOCUMENT(); # This should be the last executable line in the problem.
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