Question

设某质点受变力F=(y,x^2)作用,没曲线L:y=x^a,(a>1) 从点(0,0)移动到点(1,1),若要使F做功最

Answer 1

w=∫ydx+x^2dy=∫(从0到1) [x^a+a*x^(a+1)]dx=(a^2+2a+2)/[(a+1)(a+2)]=1-1/(a+2/a-3)

根据均值不等式,a+2/a>=2根号下（a*2/a）=2乘以根号2,取等号时a=根号2

综上,a=根号2时做功最小