Differentiable function with a constant ratio

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Differentiable function with a constant ratio

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Source: Ecuador National Olympiad OMEC level U 2024 P2 Day 1

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KAME06

#1 h Apr 5, 2025, 8:13 PM • 1 Y

Y by teomihai

Let $\alpha >0$ a real number. Given a differentiable function $f: \mathbb{R^+} \rightarrow \mathbb{R^+}$ , let $\gamma$ the curve $y=f(x)$ on the XY-plane. For all point $P$ on $\gamma$ , the tangent to $\gamma$ on $P$ intersect the x-axis and the y-axis on $A$ and $B$ , respectively, such $P \in \overline{AB}$ and $\frac{BP}{PA}=\alpha$ .
If $(20,24)$ belongs to $\gamma$ , find all possible functions $f(x)$ .

This post has been edited 1 time. Last edited by KAME06, Apr 8, 2025, 5:07 AM

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#2 h Apr 6, 2025, 10:26 AM • 1 Y

Y by teomihai

Let $P(t,f(t))$ be a point on $\gamma: y=f(x)$ such that the tangent line at $P$ computes to:

$y-f(t)=f'(t)(x-t) \Rightarrow y =f'(t) \cdot x + [f(t)-tf'(t)]$ (i);

of which the $x$ and $y-$ intercepts are: $A\left(t-\frac{f(t)}{f'(t)}, 0\right); B(0,f(t)-tf'(t))$ (ii). If $\frac{BP}{PA} = \alpha$ for $\alpha \in \mathbb{R}^{+}$ , then:

$\sqrt{t^2+t^{2}f'(y)^{2}} = \alpha \cdot \sqrt{f(t)^{2}+\frac{f(t)^{2}}{f'(t)^{2}}}$ ;

or $t^2f'(t)^{2} + t^2f'(t)^{4} = \alpha[f(t)^{2}f'(t)^{2} + f(t)^{2}]$ ;

or $f'(t)^{2} = \frac{[\alpha^{2}f(t)^{2}-t^2] \pm \sqrt{[\alpha^{2}f(t)^{2}-t^2]^2 + 4[\alpha t f(t)]^2}}{2t^2}$ ;

or $f'(t)^{2} = \frac{[\alpha^{2}f(t)^{2}-t^2] \pm \sqrt{[t^2+\alpha^{2}f(t)^{2}]^2}}{2t^2} = \frac{[\alpha^{2}f(t)^{2}-t^2] \pm [t^2+\alpha^{2}f(t)^{2}]}{2t^2}$ ;

or $f'(t)^2 = \frac{\alpha^{2}f(t)^2}{t^2}, -1 \Rightarrow f'(t) = \pm \frac{\alpha f(t)}{t} \Rightarrow \ln f(t) = \pm \alpha \ln(t) + C$ (iii).

If $f(20)=24$ is our initial condition, then we obtain:

$\ln(24) = \pm \alpha ln(20) + C \Rightarrow C = \ln\left(\frac{24}{20^{\pm \alpha}}\right)$ (iv);

which finally yields the pair of functions: $\textcolor{red}{f(t) = 24\left(\frac{t}{20}\right)^{\alpha}}$ and $\textcolor{red}{f(t) = 24\left(\frac{20}{t}\right)^{\alpha}}$ .

This post has been edited 1 time. Last edited by Mathzeus1024, Apr 6, 2025, 10:28 AM

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