试证:方程z^7-z^3+12=0的根全在圆环z绝对值大于1和小于2内

Question

试证:方程z^7-z^3+12=0的根全在圆环z绝对值大于1和小于2内

Answer 1

f(z)=z^7-z^3+12
f'(z)=7z^6-3z^2
=z^2(7z^4-3)
=z^2(√7z^2-√3)(√7z^2+√3)
|z|^2>√3/√7,f'(z)>0
f(1)0
存在解
0