The extreme values of 4 (x^2 ) ( 3 +x^2 ) ( 3 -x^2 ) over R , are

Step-by-step solution

Let aligned f(x)= & 4 (x^2 ) ( 3 +x^2 ) ( 3 -x^2 ) = & 2 (x^2 ) [ ( 2 3 )+ (2 x^2 ) ] & [ 2 A B= (A+B)+ (A-B)] = & 2 (x^2 ) [- 1 2 + (2 x^2 ) ] = & - (x^2 )+2 (x^2 ) (2 x^2 ) = & - (x^2 )+ (3 x^2 )+ (x^2 ) aligned On differentiating w.r.t. x , we get f^ (x)=- (3 x^2 )(6 x) For extremum, put f^ (x)=0 array rlrl & & - (3 x^2 )(6 x) & =0 & x & =0, array Put x=0, , in equation (i), we get aligned f(0) & = (0)=1 f( ) & = ( )=-1 aligned