Question

The number of solutions of the equation $|\sin(\pi x)| = \frac{1}{50}(x^2 + 1)$ in $\mathbb{R}$ is

• A. 28
• B. 26
• C. 14
• D. 13

Hint:

Plotting a rough graph of the parabolic curve intersecting the absolute sine humps makes this counting problem very visual.
For each full period of the sine function that lies below the maximum bound ($y \le 1$), there are exactly two intersection points.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the total number of real solutions to the transcendental equation.
Since both functions are even, the graph is symmetric about the $y$-axis. We can count the solutions for $x \gt 0$ and double the result.

Step 2: Key Formula or Approach:
- Use the range of the sine function: $|\sin(\pi x)| \le 1$.
- Determine the domain boundary where a solution is possible: $\frac{1}{50}(x^2 + 1) \le 1$.

Step 3: Detailed Explanation:

• First, we establish the range of possible solutions. Since $|\sin(\pi x)| \le 1$, we must have:
\[ \frac{1}{50}(x^2 + 1) \le 1 \implies x^2 + 1 \le 50 \implies x^2 \le 49 \implies x \in [-7, 7] \]

• At $x = 0$, the left-hand side is $|\sin(0)| = 0$, and the right-hand side is $\frac{1}{50} = 0.02$. Since $0 \neq 0.02$, $x = 0$ is not a solution.

• Let us analyze positive solutions in the interval $(0, 7]$.
The function $|\sin(\pi x)|$ is periodic with period 1. It consists of 7 "humps" on the interval $(0, 7]$, one on each interval $[k, k+1]$ for $k = 0, 1, 2, 3, 4, 5, 6$.
On each interval $[k, k+1]$:
- At the endpoints $x = k$ and $x = k+1$, the function $|\sin(\pi x)| = 0$.
- At the midpoint $x = k + 0.5$, the function reaches its peak value of 1.
- The quadratic function $y = \frac{1}{50}(x^2 + 1)$ is strictly increasing for $x \gt 0$.
- Let us compare the peak value and endpoint values of both functions:
For any $k \in \{0, 1, \dots, 6\}$, the maximum value of $y(x)$ on $[k, k+1]$ occurs at $x = k+1$, and since $x \le 7$, we have $y(k+1) \le y(7) = \frac{49+1}{50} = 1$.
Specifically, at the midpoint $x = k + 0.5$, the value of the quadratic function is:
\[ y(k+0.5) \lt y(7) = 1 \]
Since $|\sin(\pi x)|$ rises to 1 at the midpoint and falls to 0 at the endpoints, while the continuous function $y(x)$ is strictly less than 1 at the midpoint and greater than 0 at the endpoints, the two curves must intersect exactly twice in each of the 7 intervals $[k, k+1]$.

• Thus, the number of positive solutions is:
\[ 7 \times 2 = 14 \]

• Since the equation is symmetric (both sides are even functions), there are also exactly 14 negative solutions.

• The total number of solutions is:
\[ 14 + 14 = 28 \]



Step 4: Final Answer:
The number of solutions of the equation is 28.