Question

1. 函数( y = xsin x )的部分图象是（）

a.
b.
c.
d.

Explanation:

First, check the parity of \(y = x\sin x\): substitute \(x\) with \(-x\), we get \(y(-x) = (-x)\sin(-x) = (-x)(-\sin x) = x\sin x = y(x)\), so it is an even function, and its graph is symmetric about the y-axis, thus eliminating options B and D (which are asymmetric about the y-axis). Then, take a specific value: when \(x = \frac{\pi}{2}\), \(y = \frac{\pi}{2}\sin\frac{\pi}{2} = \frac{\pi}{2} > 0\), so the function value at \(x = \frac{\pi}{2}\) is positive, eliminating option C (where the value at positive \(x\) near the origin is negative). Therefore, the correct answer is A.

Answer:

A. The graph symmetric about the y-axis with positive values at positive x near the origin