Question

If x is real then the minimum value of
x2 + x +1/x2+x+1
a) 1/3
b) 3
c) 1/2
d) 1​

Answer 1

option d is the correct answer

Answer 2

Step-by-step explanation:

y=

x

2

+x+1

x

2

−x+1

yx

2

+yx+y=x

2

−x+1

⇒x

2

(y−1)+x(y+1)+(y−1)=0

x is real, so D≥0,

⟹(y+1)

2

−4(y−1)

2

≥0

⟹y

2

+1+2y−4[y

2

+1−2y]≥0

⟹y

2

+1+2y−4y

2

−4+8y≥0

⟹−3y

2

+10y−3≥0

⟹3y

2

−10y+3≤0

⟹3y

2

−9y−y+3≤0

⟹3y(y−3)−(y−3)≤0

⟹(3y−1)(y−3)≤0

⟹y≤3 and y≥

3

1

⟹

3

1

≤y≤3

Hence,

3

1

is the minimum value of y.