θ为锐角,求(sinθ)^6+(cosθ)^6的最小值.

ataorj

θ为锐角,求(sinθ)^6+(cosθ)^6的最小值.
解:[这是很简单的初中题]
设a=sinθ,b=cosθ,则
a^6+b^6
=(a^2+b^2)(a^4+b^4-a^2b^2)
=1*((a^2+b^2)^2-3a^2b^2)
=1-3a^2b^2
=1-3a^2(1-a^2)
=1-3a^2+3a^4
其当a^2=-(-3)/2/3=0.5时有最小值(4*3*1-3^2)/4/3=0.25
a^2=0.5表明θ=45°