Question

Let $f(x) = 4x(1-x), 0 \leq x \leq 1$. The number of solution of $f(f(f(x))) = \frac{x}{3}$ is:

• A. 2
• B. 4
• C. 8
• D. 16

Answer 1

Answer: (C)

8

Answer 2

Let $f(x) = 4x(1-x)$. We want to find the number of solutions for $f(f(f(x))) = \frac{x}{3}$ for $0 \leq x \leq 1$.

Step 1: Use trigonometric substitution.

Let $x = \sin^2 \theta$. Since $0 \leq x \leq 1$, we can choose $\theta \in [0, \pi/2]$. Each $x$ corresponds to a unique $\theta$.

Substitute $x = \sin^2 \theta$ into $f(x)$: $f(x) = 4\sin^2 \theta (1-\sin^2 \theta) = 4\sin^2 \theta \cos^2 \theta = (2\sin\theta\cos\theta)^2 = (\sin 2\theta)^2$.

Applying this iteratively: $f(f(x)) = f((\sin 2\theta)^2) = (\sin(2 \cdot 2\theta))^2 = (\sin 4\theta)^2$. $f(f(f(x))) = f((\sin 4\theta)^2) = (\sin(2 \cdot 4\theta))^2 = (\sin 8\theta)^2$.

Step 2: Rewrite the equation in terms of $\theta$.

The equation becomes $(\sin 8\theta)^2 = \frac{\sin^2 \theta}{3}$. This implies $3\sin^2 8\theta = \sin^2 \theta$. Taking the square root of both sides: $\sqrt{3}|\sin 8\theta| = |\sin \theta|$. Since $\theta \in [0, \pi/2]$, $\sin \theta \geq 0$, so $|\sin \theta| = \sin \theta$. Thus, $\sqrt{3}|\sin 8\theta| = \sin \theta$. This leads to two cases:

Case 1: $\sqrt{3}\sin 8\theta = \sin \theta$ Case 2: $\sqrt{3}\sin 8\theta = -\sin \theta$

Step 3: Analyze the solution $x=0$.

If $x=0$, then $\theta=0$. $f(f(f(0))) = 0$ and $0/3 = 0$. So $x=0$ is a solution. In terms of $\theta$, $\sin 8(0) = 0$ and $\sin 0 = 0$, so $\sqrt{3}|0| = 0$, which is true.

Step 4: Analyze solutions for $\theta \in (0, \pi/2)$.

For $\theta \in (0, \pi/2)$, $\sin \theta > 0$, so we can divide by $\sin \theta$. Let $g(\theta) = \frac{\sin 8\theta}{\sin \theta}$. The equations become $g(\theta) = \frac{1}{\sqrt{3}}$ and $g(\theta) = -\frac{1}{\sqrt{3}}$. We know that $g(\theta) = U_7(\cos \theta)$, where $U_7$ is the Chebyshev polynomial of the second kind. Let's analyze the graph of $g(\theta)$ for $\theta \in (0, \pi/2)$. As $\theta \to 0^+$, $g(\theta) \to 8$. The roots of $g(\theta)=0$ in $(0, \pi/2)$ are $8\theta = k\pi \implies \theta = k\pi/8$ for $k=1,2,3$. So $\theta = \pi/8, \pi/4, 3\pi/8$. The local extrema of $g(\theta)$ in $(0, \pi/2)$ occur at $\theta = (2k-1)\pi/16$ for $k=1,2,3,4$. $g(\pi/16) = \frac{\sin(\pi/2)}{\sin(\pi/16)} = \frac{1}{\sin(\pi/16)} \approx 5.12$ (local maximum). $g(3\pi/16) = \frac{\sin(3\pi/2)}{\sin(3\pi/16)} = \frac{-1}{\sin(3\pi/16)} \approx -1.8$ (local minimum). $g(5\pi/16) = \frac{\sin(5\pi/2)}{\sin(5\pi/16)} = \frac{1}{\sin(5\pi/16)} \approx 1.2$ (local maximum). $g(7\pi/16) = \frac{\sin(7\pi/2)}{\sin(7\pi/16)} = \frac{-1}{\sin(7\pi/16)} \approx -1.02$ (local minimum). At $\theta=\pi/2$, $g(\pi/2)=0$.

Let $C_1 = 1/\sqrt{3} \approx 0.577$ and $C_2 = -1/\sqrt{3} \approx -0.577$.

Counting solutions for $g(\theta) = C_1$:

• In $(0, \pi/8)$: $g(\theta)$ starts at 8, decreases to $g(\pi/16) \approx 5.12$, then decreases to 0. It crosses $C_1 \approx 0.577$ once (between $\pi/16$ and $\pi/8$). (1 solution)
• In $(\pi/8, \pi/4)$: $g(\theta)$ is negative. No solutions.
• In $(\pi/4, 3\pi/8)$: $g(\theta)$ goes from 0, increases to $g(5\pi/16) \approx 1.2$, then decreases to 0. It crosses $C_1$ twice. (2 solutions)
• In $(3\pi/8, \pi/2)$: $g(\theta)$ is negative. No solutions. Total solutions for $g(\theta) = C_1$ in $(0, \pi/2)$ is $1+2=3$.

Counting solutions for $g(\theta) = C_2$:

• In $(0, \pi/8)$: $g(\theta)$ is positive. No solutions.
• In $(\pi/8, \pi/4)$: $g(\theta)$ goes from 0, decreases to $g(3\pi/16) \approx -1.8$, then increases to 0. It crosses $C_2 \approx -0.577$ twice. (2 solutions)
• In $(\pi/4, 3\pi/8)$: $g(\theta)$ is positive. No solutions.
• In $(3\pi/8, \pi/2)$: $g(\theta)$ goes from 0, decreases to $g(7\pi/16) \approx -1.02$, then increases to 0. It crosses $C_2$ twice. (2 solutions) Total solutions for $g(\theta) = C_2$ in $(0, \pi/2)$ is $2+2=4$.

Step 5: Total number of solutions.

The total number of solutions for $\theta \in (0, \pi/2)$ is $3+4=7$. Each of these $\theta$ values corresponds to a unique $x \in (0,1)$. Adding the solution $x=0$ (corresponding to $\theta=0$), the total number of solutions for $x$ is $7+1=8$.

The final answer is $\boxed{\text{8}}$.