Question

高等数学，这个函数是增函数还是减函数

f(x)=x^0.25 - x^0.5 -cos x
定义域（0,+∞）

Answer 1

f(x)=x^0.25-x^0.5-cosx

f'(x)=0.25x^(-0.75)-0.5x^(-0.5)+sinx

令g(x)=0.25x^(-0.75)-0.5x^(-0.5)

g^' (x)=-0.1875x^(-1.75)+0.25x^(-1.5)

极值点g'(x)=0

-0.1875x^(-1.75)+0.25x^(-1.5)=0

-0.1875x^(-1.75)•x^1.75+0.25x^(-1.5)•x^1.75=0

0.25x^0.25=0.1875

x=0.75^4

 gmin (x)=g(0.75^4 )>-0.3

lim(n→+∞) (0.25x^(-0.75)-0.5x^(-0.5))=0

∴-0.3<g(x)<0

-1.3<f^' (x)<1

即f(x)在(0,+∞)内增减区间随sinx变化而呈现类似周期性的变化

Answer 2

减函数