1993 Paper 1 Q7

Year: 1993
Paper: 1
Question Number: 7

Course: LFM Stats And Pure
Section: Curve Sketching

Difficulty: 1500.0 Banger: 1516.0

curve sketching cubic polynomial stationary points discriminant inequality conditions for real roots turning points

Problem

Sketch the curve \[ \mathrm{f}(x)=x^{3}+Ax^{2}+B \] first in the case \(A>0\) and \(B>0\), and then in the case \(A<0\) and \(B>0.\) Show that the equation \[ x^{3}+ax^{2}+b=0, \] where \(a\) and \(b\) are real, will have three distinct real roots if \[ 27b^{2}+3a^{3}b<0, \] but will have fewer than three if \[ 27b^{2}+4a^{3}b<0. \]

No solution available for this problem.

Rating Information

Difficulty Rating: 1500.0

Difficulty Comparisons: 0

Banger Rating: 1516.0

Banger Comparisons: 1

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Problem source

Sketch the curve 
\[
\mathrm{f}(x)=x^{3}+Ax^{2}+B
\]
first in the case $A>0$ and $B>0$, and then in the case $A<0$ and
$B>0.$

Show that the equation 
\[
x^{3}+ax^{2}+b=0,
\]
where $a$ and $b$ are real, will have three distinct real roots
if 
\[
27b^{2}+3a^{3}b<0,
\]
but will have fewer than three if 
\[
27b^{2}+4a^{3}b<0.
\]

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