Question

1. 函数$y = xsin x$的部分图象是（）

a.
!img(https://p3-hippo-sign.byteimg.com/tos-cn-i-a9yeduch1e/78c8c01c842d4b469c2c8c8c8c8c8c8~tplv-ii0cwwkcx9-resize-crop-v1:60:160:240:380:0:0.png?lk3s=f89d68d8&x-expires=2044327375&x-signature=e8c8c8c8c8c8c8c8c8c8c8c8c8c8c8c8)
b.
!img(https://p3-hippo-sign.byteimg.com/tos-cn-i-a9yeduch1e/78c8c01c842d4b469c2c8c8c8c8c8c8~tplv-ii0cwwkcx9-resize-crop-v1:320:160:500:380:0:0.png?lk3s=f89d68d8&x-expires=2044327375&x-signature=e8c8c8c8c8c8c8c8c8c8c8c8c8c8c8c8)
c.
!img(https://p3-hippo-sign.byteimg.com/tos-cn-i-a9yeduch1e/78c8c01c842d4b469c2c8c8c8c8c8c8~tplv-ii0cwwkcx9-resize-crop-v1:60:480:240:700:0:0.png?lk3s=f89d68d8&x-expires=2044327375&x-signature=e8c8c8c8c8c8c8c8c8c8c8c8c8c8c8c8)
d.
!img(https://p3-hippo-sign.byteimg.com/tos-cn-i-a9yeduch1e/78c8c01c842d4b469c2c8c8c8c8c8c8~tplv-ii0cwwkcx9-resize-crop-v1:320:480:500:700:0:0.png?lk3s=f89d68d8&x-expires=2044327375&x-signature=e8c8c8c8c8c8c8c8c8c8c8c8c8c8c8c8)

Explanation:

Step1: 判断函数的奇偶性

对于函数\(y = f(x)=x\sin x\)，其定义域为\(R\)，关于原点对称。计算\(f(-x)\)：\(f(-x)=(-x)\sin(-x)\)，根据正弦函数的性质\(\sin(-x)=-\sin x\)，则\(f(-x)=(-x)\times(-\sin x)=x\sin x = f(x)\)，所以函数\(y = x\sin x\)是偶函数，其图象关于\(y\)轴对称，因此可以排除选项B和D，因为B和D的图象不关于\(y\)轴对称。

Step2: 分析函数在\(x\in(0,\pi)\)时的取值

当\(x\in(0,\pi)\)时，\(x>0\)，且\(\sin x>0\)（因为正弦函数在\((0,\pi)\)上的值大于\(0\)），根据正数乘以正数为正数，所以\(y = x\sin x>0\)。观察选项A和C，选项A在\(x\in(0,\pi)\)时，函数值\(y>0\)；选项C在\(x\in(0,\pi)\)时，函数值\(y<0\)，所以排除选项C。

Answer:

A