What is the minimum value of + 2 ?

Step-by-step solution

Let A = + 2 On differentiating w.r.t. to , we get d A d =- -2 2 Put dA d =0 for maximaor minima. +2 2 =0 +4 (1+4 )=0 =0 , or 4 +1=0 =1 or =- 1 4 Now, d ² ~A ~d ² =- -4 2 =- -4 (2 ² -1 ) For =1 d ² ~A ~d ² =- -4 (2 ² -1 )=-1-4(2(1)-1)=-1-4=-5 < 0 So, A is maximum at =1 ( d ² ~A ~d ² )_ = -1 4 = 1 4 -4 (2 1 16 -1 )>0 [ . Since . 2 =2 ² -1 ] A is minimum at = ⁻¹ ( -1 4 ) Now minimum value of + 2 or of +2 ² -1= ( -1 4 )+2 ( 1 16 )-1= -1 4 + 1 8 -1= -2+1-8 8 = -9 8