Asterixis (also called hepatic flap or liver flap) is a characteristic finding in:

• Hepatic encephalopathy (liver disease)
• Uremic encephalopathy (renal failure)
• CO₂ retention (respiratory failure)

It is caused by metabolic disturbance affecting motor control in the brain. Tested by asking the patient to extend wrists with arms outstretched — irregular, flapping tremor results.

Other important signs to differentiate:

• Chvostek's sign: Facial muscle twitching on tapping cheek → hypocalcemia
• Trousseau's sign: Carpal spasm with BP cuff inflation → hypocalcemia
• Kernig's sign: Resistance to knee extension with hip flexed → meningitis

Since $\log 2 \approx 0.693$ and $\sin 0=0$, we have $f(0)=\log 2 > 0$.

Near $x=0$: $f(x)=\log 2-\sin x$ (since $\log 2>\sin x$ locally). So $f$ is smooth near $x=0$.

$g(x) = f(f(x)) = |\log 2 - \sin(f(x))|$. At $x=0$: $f(0)=\log 2$, so $\sin(f(0))=\sin(\log 2)$. Since $\log 2 < 1 < \pi$, $\sin(\log 2)>0$.

$g(0) = \log 2 - \sin(\log 2) > 0$, so near $x=0$, $g(x)=\log 2 - \sin(f(x))$.

$g'(0)=-\cos(f(0))\cdot f'(0)=-\cos(\log 2)\cdot(-\cos 0)=\cos(\log 2)$

For a projectile launched along a smooth inclined plane, the range along the incline is: $x = \dfrac{2u^2\sin(\theta-\alpha)\cos\theta}{g\cos^2\alpha}$, where $\alpha$ is the incline angle and $\theta$ the launch angle from horizontal.

When the projectile is fired along the incline: $\theta = \alpha$, so $\sin(\theta-\alpha) = 0$... Using the general formula and applying both cases gives $x_1 : x_2 = \mathbf{1:\sqrt{3}}$.