令t=x^(1/3),则
h(x)=x^(7/3)+x^(4/3)-3*x^(1/3)
=t^7+t^4-3t
=g(t)
求h(x)的极值相当于求g(t)的极值
g'(t)=7t^6+4t^3-3=(t^3+1)(7t^3-3)
令g'(t)=0可解得
t^3=-1或t^3=3/7
易验证,t^3≤-1或t^3≥3/7时,g'(t)≥0
-1≤t^3≤3/7时,g'(t)≤0
∴g(t)在t^3=-1,即t=-1处取得极大值,g(-1)=-1+1+3=2
g(t)在t^3=3/7,即t=(3/7)^(1/3)处取得极小值,
极小值为 g[(3/7)^(1/3)]=(3/7)^(7/3)+(3/7)^(4/3)-3*(3/7)^(1/3)
此极大极小值与h(x)的极大极小值相同