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20 15 Fujian mathematics quality inspection multiple-choice questions. Question 10. Important steps
10x & gt; 0 xf'(x)-f(x)=x,①

∴f'(x)= 1+f(x)/x,

f''(x)=[xf'(x)-f(x)]/x^2= 1/x,

∴f'(x)=lnx+b,

f( 1)= 1,f'( 1)=2=b,

f'(x)=lnx+2,

f(x)=xlnx+x+c,

∴ 1= 1+c,

∴c=0,

∴f(x)=xlnx+x.

① ② ③ Correct,

Let g (x) = f (x)-x 2, g' (x) = lnx+2-2x, g'' (x) = 1/x-2,

0<x & ltG'' (x) at 1/2 >: 0, g'(x) is increasing function, others, g'(x) is a decreasing function,

∴g'(x)<; =g'( 1/2)= 1-ln2,g'( 1)=0,

0<x & ltG' (x) at 1 >: 0, g(x) is increasing function; Otherwise, g(x) is a decreasing function,

∴g(x)<; =g( 1)=0,

4 established.

Choose D.