If and are the least and the greatest values of f(x)= ( ⁻¹ x )^2+ ( ⁻¹ x )^2 for all x R respectively, then 8( + )=

Step-by-step solution

f(x)= ( ⁻¹ x )^2+ ( ⁻¹ x )^2 Let ⁻¹ x=a and ⁻¹ x=b Then, aligned f(x) & =a^2+b^2 & =(a+b)^2-2 a b aligned Put the value of a and b aligned & f(x)= ( ⁻¹ x+ ⁻¹ x )^2-2 ⁻¹ x ⁻¹ x & = ^2 4 -2 ⁻¹ x ⁻¹ x [ ⁻¹ x+ ⁻¹ x= / 2 ] & = ^2 4 -2 ⁻¹ x ( 2 - ⁻¹ x ) & = ^2 4 - ⁻¹ x+2 ( ⁻¹ x )^2 aligned For minimum and maximum value, aligned f^ (x) & =0- 1 1-x^2 +4 ⁻¹ x 1 1-x^2 =0 & = 1 1-x^2 [4 ⁻¹ x- ]=0= ⁻¹ x= / 4 x & = / 4= 1 2 aligned Therefore, f^ ( 1 2 )=+ ve aligned & f(x)_ = when x=1 / 2 & f(x)_ = ^2 4 -2 ⁻¹ x ⁻¹ x & f(x)_ = ^