A cubic that is increasing for every x
A cubic that is increasing for every x
IP 4
8 marks
The function \(\mathrm f\) is defined by \(\mathrm f(x)=x^3-3x^2+3x-7\) for all real \(x\).
- Show that \(\mathrm f'(x)=3(x-1)^2\), and explain why \(\mathrm f\) is increasing for all real \(x\).[3]
- Hence explain why the equation \(\mathrm f(x)=0\) has exactly one real root, and show that this root lies between \(2\) and \(3\).[3]
- By writing \(\mathrm f(x)\) in the form \((x-a)^3+b\), find the exact root of \(\mathrm f(x)=0\).[2]
Solution:
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Answer:(a) shown; \(\mathrm f'(x)\ge0\) with equality only at \(x=1\) (b) at most one root since \(\mathrm f\) is increasing; \(\mathrm f(2)=-5\), \(\mathrm f(3)=2\) (c) \(x=1+\sqrt[3]{6}\)
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