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Question

For all real x, the minimum value of 1x+x21+x+x2 is

A
13
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B
1
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C
3
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Solution

The correct option is A 13

Let z=1x+x21+x+x2
z+zx+zx2=1x+x2
zx2x2+zx+x+z1=0
x2(z1)+x(z+1)+(z1)=0
For real x, B24AC0
(z+1)24(z1)(z1)0
z2+2z+14z2+8z40
3z2+10z303z2+9z+z30
3z(z3)+1(z3)0
(z3)(3z+1)013z3
minimum value of z=13


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