【正确答案】

【答案解析】2(x ² +y ² )f(xy) [解析] z=∫ ¹ ₀ |xy-t|f(t)dt
=∫ ^xy ₀ (xy-t)f(t)dt+∫ ¹ _xy (t-xy)f(t)dt
=xy∫ ₀ ^xy f(t)dt-∫ ₀ ^xy (t)dt+∫ ¹ _xy tf(t)dt-xy∫ ¹ _xy (t)dt则 z" _x =y∫ ₀ ^xy f(t)dt+xy ² f(xy)-xy ² f(xy)-xy ² f(xy)-y∫ ¹ _xy f(t)dt+xy ² f(xy)
=y∫ ₀ ^xy f(t)dt-y∫ ¹ _xy f(t)dt
z" _xx =y ² f(xy)+y ² f(xy)=2y ² f(xy)
由变量对称性知
z" _yy =2x ² f(xy)则 z" _xx +z" _yy =2(x ² +y ² )f(xy)．