The maximum value of y=x( x)^2 is

Step-by-step solution

y=x( x)^2 For maximum value aligned d y d x & =x (2 x 1 x )+( x)^2 & =2 x+( x)^2 aligned Now, put d y d x =0 aligned & 2 x+( x)^2=0 & ( x)[ x+2]=0 & x=0 x+2=0 & x=1 & x=-2 & x=e⁻² & aligned Using second order derivative test. Again differentiating y w.r.t. x aligned & d^2 y d x^2 = 2 x +2 x 1 x & . d^2 y d x^2 |_ x=1 =2>0 aligned x=1 is point of minima aligned . d^2 y d x^2 |_ x=e⁻² & = 2 e⁻² +2 e⁻² 1 e⁻² & = 1 e⁻² (2-4)=- 2 e⁻² < 0 y_ & =e⁻² ( e⁻² )^2 & =e⁻²(-2)^2=4 e⁻² aligned