Question

1. 函数$y = xsin x$的部分图象是（）

a.
!a选项图像(https://via.placeholder.com/150)
b.
!b选项图像(https://via.placeholder.com/150)
c.
!c选项图像(https://via.placeholder.com/150)
d.
!d选项图像(https://via.placeholder.com/150)

Explanation:

Step1: Analyze the parity of the function

To determine the parity of the function \( y = x\sin x \), we substitute \( -x \) into the function:
\( f(-x) = (-x)\sin(-x) \)
Since \( \sin(-x)=-\sin x \), we have:
\( f(-x)=(-x)(-\sin x)=x\sin x = f(x) \)
So the function \( y = x\sin x \) is an even function, and its graph is symmetric about the \( y \)-axis. This eliminates options B and D because their graphs are not symmetric about the \( y \)-axis.

Step2: Analyze the function value in the interval \( (0, \pi) \)

When \( x\in(0, \pi) \), \( \sin x>0 \) and \( x > 0 \), so \( y=x\sin x>0 \).
For option C, in the interval \( (0, \pi) \), the function value is negative (the graph is below the \( x \)-axis), which does not match.
For option A, in the interval \( (0, \pi) \), the function value is positive (the graph is above the \( x \)-axis), which matches our analysis.

Answer:

A