أثبت أنه لا يمكن أن يكون للمعادلة س4 +2س2 –4 = 0 أكثر من جذرين حقيقيين مختلفين .

 

 

let f (x)=x^4+2x^2

f is continuous and differentiable on R SO IT IS ON ANY CLOSED INTERVAL

let a,b,c are three different zeroes for the function then f(a)=0


f(b)=0 ,f(c)=0




if we apply rolle,s theorem then we have two numbers d , e where d is between a,b and e is between b,c where f prime (d) =0 and f prime (e) =0 f pime means the derivative o f f


but f prime (x)=4 x^3 - 4x and 4x^3 -4x =0 when x =0 and this contradicts what we say before so the function can not have more than two different 2 raal zeroes

 

 

[IMG][/IMG]

 

 

عفوا الفترة ليست )0و1( وانما )1و2(