導(dǎo)數(shù)問(wèn)題9設(shè)函數(shù)fn(x)=n^2*x^2*(1-x)^n（n為正整數(shù)），則fn(x)在[0,1]上的最大值為答案4(n/n+2)^(n+1)

熱心網(wǎng)友

fn(x)=n^2*x^2*(1-x)^nfn'(x)=2*n^2*x*(1-x)^n+n^2*x^2*(-n)*(1-x)^(n-1)=n^2*x*(1-x)^(n-1)*[2-(n+2)x]駐點(diǎn)：x=0（舍去）,x=1（舍去）,x=2/(n+2)∈（0，1）fn(0)=0,fn(1)=0,fn[2/(n+2)]=4*[n/(n+2)]^(n+2)0所以這個(gè)函數(shù)在[0，1]上的最大值為：fn[2/(n+2)]=4*[n/(n+2)]^(n+2)。