Constants of a quadratic and an exponential function whose graphs share a tangent at a given point
Constants of a quadratic and an exponential function whose graphs share a tangent at a given point
7 marks
The functions \(f\) and \(g\) are given by \(f(x)=(x-a)^2-3b\) and \(g(x)=2\mathrm{e}^{x-1}-b\), where \(a\) and \(b\) are real constants.
- Write down an expression for \(f'(x)\).[1]
At the point where \(x=2\), the graphs of \(f\) and \(g\) touch and have the same tangent line.
- Show that \(a=2-\mathrm{e}\).[3]
- Hence, show that \(b=\dfrac{\mathrm{e}(\mathrm{e}-2)}{2}\).[3]
Solution:
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Answer:(a) \(f'(x)=2(x-a)\) (b) \(2(2-a)=2\mathrm{e}\Rightarrow a=2-\mathrm{e}\) (c) \(\mathrm{e}^2-3b=2\mathrm{e}-b\Rightarrow b=\dfrac{\mathrm{e}(\mathrm{e}-2)}{2}\)
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