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The functions f and g are defined by

\begin{aligned} & f(x)=\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{2}, x \in \mathbb{R} \\ & g(x)=\frac{\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}, x \in \mathbb{R} \end{aligned}

(a)

Let h(x)=n f(x)+g(x) where $n \in \mathbb{R}, n>1$ .

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(i)

By forming a quadratic equation in $\mathrm{e}^{x}$ , solve the equation h(x)=k, where $k \in \mathbb{R}^{+}$ .

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$\quad(n+1) \mathrm{e}^{2 x}-2 k \mathrm{e}^{x}+(n-1)=0$ $\mathrm{e}^{x}=\frac{2 k \pm \sqrt{4 k^{2}-4\left(n^{2}-1\right)}}{2(n+1)}$ $x=\ln \left(\frac{k \pm \sqrt{k^{2}-n^{2}+1}}{n+1}\right)$

(ii)

Hence or otherwise show that the equation h(x)=k has two real solutions provided that $k>\sqrt{n^{2}-1}$ and $k \in \mathbb{R}^{+}$ .

[ 3 ]

for two real solutions, we require $k>\sqrt{k^{2}-n^{2}+1}$
and we also require $k^{2}-n^{2}+1>0$ $k^{2}>n^{2}-1$ $\Rightarrow k>\sqrt{n^{2}-1}\left(k \in \mathbb{R}^{+}\right)$

IB Maths AA HL Sl 2 7 Quadratic Equations and Inequalities Topic Practice

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IB Maths AA HL Sl 2 7 Quadratic Equations and Inequalities Topic Practice

Learn this topic

Question 1

Question (a)

Question (i)

Question (ii)