Stationary Points & Inflection Points
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Find any points of inflection of the following curves:
(a)\(f(x)=x^3-6x^2+12x-1\)
(b)\(f(x)=x^4-6x^2+4\)
(c)\(f(x)=xe^{-x}\)
(a)
\(f''(x)=6x-12=0\Rightarrow x=2\)
Check sign change: \(f''(1)=-6<0,\;f''(3)=6>0\) ✓
Answer
Inflection at \((2,7)\)
(b)
\(f''(x)=12x^2-12=12(x^2-1)=0\Rightarrow x=\pm1\)
Check sign changes: both valid
Answer
Inflections at \((1,-1)\) and \((-1,-1)\)
(c)
\(f'=e^{-x}-xe^{-x},\;f''=-e^{-x}-e^{-x}+xe^{-x}=e^{-x}(x-2)=0\Rightarrow x=2\)
Answer
Inflection at \((2,2e^{-2})\)
Question 2
For \(f(x)=\dfrac{x^3}{3}-x^2-3x+2\), find all stationary points and classify them, and find any inflection points.
First derivative
\(f'(x)=x^2-2x-3=(x-3)(x+1)=0\Rightarrow x=3,-1\)
Second derivative
\(f''(x)=2x-2\); \(f''(-1)=-4<0\) (max); \(f''(3)=4>0\) (min)
Inflection
\(f''=0\Rightarrow x=1\); \(f(1)=\tfrac{1}{3}-1-3+2=-\tfrac{5}{3}\)
Answer
Local max \((-1,\tfrac{11}{3})\); local min \((3,-7)\); inflection \((1,-\tfrac{5}{3})\)
Question 3
Show that \(f(x)=x^4\) has a stationary point but no inflection point (where \(f''=0\) is not a sign change).
First derivative
\(f'(x)=4x^3=0\Rightarrow x=0\) (minimum, by the first derivative test: \(f'<0\) for \(x<0\), \(f'>0\) for \(x>0\))
Second derivative
\(f''(x)=12x^2=0\Rightarrow x=0\). Check sign change: \(f''(-1)=12>0;\;f''(1)=12>0\) — no sign change (both positive), so \(x=0\) is NOT an inflection point despite \(f''(0)=0\).
Conclusion
Stationary point (minimum) at \((0,0)\); no inflection point, since \(f''\) never changes sign.
Question 4
Use the second derivative test to classify the stationary points of \(y=\dfrac{\ln x}{x}\), \(x>0\).
First derivative
\(y'=\dfrac{\tfrac{1}{x}\cdot x-\ln x}{x^2}=\dfrac{1-\ln x}{x^2}=0\Rightarrow x=e\)
Second derivative
\(y''=\dfrac{-\tfrac{1}{x}\cdot x^2-(1-\ln x)\cdot2x}{x^4}=\dfrac{-1-2+2\ln x}{x^3}=\dfrac{2\ln x-3}{x^3}\)
At \(x=e\)
\(y''(e)=\dfrac{2-3}{e^3}=-\dfrac{1}{e^3}<0\)
Answer
Local maximum at \(\left(e,\,\dfrac{1}{e}\right)\)
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