Rules for unknown functions and a false claim about squares
Rules for unknown functions and a false claim about squares
IP 4
8 marks
The functions \(\mathrm f\) and \(\mathrm g\) are differentiable for all real \(x\), with \(\mathrm f(2)=3\), \(\mathrm f'(2)=-1\), \(\mathrm g(2)=4\) and \(\mathrm g'(2)=5\). No formulas for \(\mathrm f\) and \(\mathrm g\) are given.
- Find \(\mathrm h'(2)\), where \(\mathrm h(x)=\dfrac{\mathrm f(x)\mathrm g(x)}{1+\mathrm f(x)}\).[3]
- Find \(\mathrm k'(2)\), where \(\mathrm k(x)=\mathrm f\left(x^2-2\right)\).[2]
- A student claims: "If \(\mathrm f\) is increasing for all real \(x\), then \([\mathrm f(x)]^2\) is also increasing for all real \(x\)." Give a counter-example, and prove that the claim is true when, in addition, \(\mathrm f(x)>0\) for all \(x\).[3]
Solution:
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Answer:(a) \(\mathrm h'(2)=\dfrac72\) (b) \(\mathrm k'(2)=-4\) (c) \(\mathrm f(x)=x\); with \(\mathrm f>0\), \(\dfrac{\mathrm d}{\mathrm dx}\mathrm f^2=2\mathrm f\mathrm f'\ge0\)
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