Inequality problems

---------------------

圖片參考：http://imgcld.yimg.com/8/n/HA00430051/o/701012190121913873429400.jpg


圖片參考：http://imgcld.yimg.com/8/n/HA00430051/o/701012190121913873429401.jpg

1.(b) the condition should be: ab is positive real no.
(a and b have the same sign or a and b are conjugates)

1(a) (a-b)^2=a^2-2ab+b^2 (b) As (a+b)^2 >= 0a^2-2ab+b^2 >= 0a^2+b^2 >= 2ab(a+b)^2-2ab >= 2ab(a+b)^2 >= 4ab(a+b)^2/4 >= ab(a+b)/2 >= √ab2(a) a^2=b^2+c^2-2bc(cosA). Rearranging, c^2-2bc(cosA)+b^2-a^2=0. Treat it as a quadratic equation of variable c. Then if c has two solutions, its discriminant should be > 0. [(2b)(cosA)]^2-4(b^2-a^2) > 0(4b^2)(cosA)^2 > 4(b^2-a^2)(cosA)^2 > 1-a^2/b^2...(*)(b) a=3,b=4 ; (*) becomes(cosA)^2 > 1-9/16(cosA)^2 > 7/16cosA < -√7/4 or cosA > √7/40 < A < 49 or 131 < A < 180 (because 0 < A < 180)