Geometrical Connections (HL)
IB Mathematics AA · HL · Shadow Worksheet
Practice
Question 1
The graph of \(f(x)\) has the following features: \(f'(x)>0\) for \(x<-1\) and \(x>3\); \(f'(x)<0\) for \(-10\) for \(x>1\); \(f''(x)<0\) for \(x<1\). Describe the nature of the stationary points and point of inflection.
At \(x=-1\)
\(f'\) changes from \(+\) to \(-\): local maximum
At \(x=3\)
\(f'\) changes from \(-\) to \(+\): local minimum
At \(x=1\)
\(f''\) changes sign: point of inflection
Answer
Local max at \(x=-1\); local min at \(x=3\); inflection at \(x=1\)
Question 2
Sketch \(f'(x)\) given that \(f(x)=x^3-3x^2-9x+2\).
Differentiate
\(f'(x)=3x^2-6x-9=3(x-3)(x+1)\)
Features of \(f'\)
Parabola opening up; roots at \(x=-1\) and \(x=3\); vertex at \(x=1,\;f'(1)=-12\)
Sketch
Upward parabola with \(x\)-intercepts at \(-1\) and \(3\), vertex below axis at \((1,-12)\)
Question 3
Given the graph of \(y=f'(x)\) is a straight line from \((-2,4)\) to \((4,-2)\), sketch a possible graph of \(y=f(x)\), identifying stationary points and concavity.
\(f'(x)=0\) when
Line crosses \(x\)-axis: \(\dfrac{4-0}{-2-x_0}=\dfrac{4}{-2-x_0}\Rightarrow\) solve: gradient \(= frac{-2-4}{4-(-2)}=-1\); \(y=-x+2\Rightarrow x=2\)
Concavity
\(f''>0\) where \(f'\) increasing — line has negative slope so \(f''\) is constant negative: concave down throughout
Sketch features
Local max at \(x=2\); concave down everywhere; no inflection
Question 4
For \(f(x)=x^4-8x^2+3\), find all stationary points, inflection points, and sketch.
First derivative
\(f'(x)=4x^3-16x=4x(x^2-4)=4x(x-2)(x+2)=0\Rightarrow x=0,\pm2\)
Second derivative
\(f''(x)=12x^2-16\); \(f''(0)=-16<0\) (max); \(f''(\pm2)=32>0\) (min)
Inflection: \(f''=0\)
\(12x^2=16\Rightarrow x=\pm frac{2}{\sqrt3}\)
Answer
Local max \((0,3)\); local mins \((\pm2,-13)\); inflections at \(x=\pm frac{2}{\sqrt3}\)
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