The greatest and least values of ( ⁻¹ x )²+ ( ⁻¹ x )² are respectively

Step-by-step solution

We have, ( ⁻¹ x )²+ ( ⁻¹ x )²= ( ⁻¹ x+ ⁻¹ x )²-2 ⁻¹ x ⁻¹ x= ² 4 -2 ⁻¹ x ( 2 - ⁻¹ x )= ² 4 - ⁻¹ x+2 ( ⁻¹ x )²=2 [ ( ⁻¹ x )²- 2 ⁻¹ x+ ² 8 ]=2 [ ( ⁻¹ x- 4 )²+ ² 16 ] Thus, the least value is 2 ( ² 16 ) i.e. ² 8 and the greatest value is 2 [ ( - 2 - 4 )²+ ² 16 ] i.e. 5 ² 4 .