Rolle theorem and equations of degree3

[madah12]

Homework Statement

prove that any equation
ax³+bx²+cx+d has at most 3 solutions

Homework Equations

if f is continuous on [a,b] ,f is differentiable on (a,b) and f(a)=f(b)
then there exist a<k<b such that f'(k)= 0

The Attempt at a Solution

suppose f has 4 distinct solutions a,b,c,d then f(a)=f(b)=f(c)=f(d)=0
by rolle theorem there exist a k1 between a and b where f'(k1)=0
also there exist k2 between b and c f'(k2)=0 and k3 between c and d where f'(k3)=0
and f'(x) = 3ax²+2bx+c
I know that it is supposed to have two only solutions
but how can i prove that?

 

[epenguin]

Homework Statement

prove that any equation
ax³+bx²+cx+d has at most 3 solutions

Homework Equations

if f is continuous on [a,b] ,f is differentiable on (a,b) and f(a)=f(b)
then there exist a<k<b such that f'(k)= 0

The Attempt at a Solution

suppose f has 4 distinct solutions a,b,c,d then f(a)=f(b)=f(c)=f(d)=0
by rolle theorem there exist a k1 between a and b where f'(k1)=0
also there exist k2 between b and c f'(k2)=0 and k3 between c and d where f'(k3)=0
and f'(x) = 3ax²+2bx+c
I know that it is supposed to have two only solutions
but how can i prove that?

I would do it for max 2 real solutions to the quadratic exactly the same way you are trying to prove max three for the cubic.

You can and should go on to prove the max is n for the n-ic.

(When you say 'solutions' here you are supposed to say 'real' though everyone will know what you mean.)

 

[madah12]

so f' since f' is cont. and diff. on R
suppose f' has three real distinct roots r₁ and r₂ and r₃ then f(r₁)=f(r₂)=f(r₃)=0
by rolle theorem
there exist a Z1 between r₁and r₂ such that f''(Z1) =0
and there exist Z2 between ₂and r₃such that f''(Z2)=0
but f''(x) =6ax+2b
suppose f'' has two roots r_a ,r_b
f'' is cont. and diff. on r
there exist a Q such that f'''(q)=0
but f'''(q)=6a and since f is from degree 3 a =! 0
which leads to a contradictions
Edit: @epenguin yeah sorry actually in the first day my instructor wrote in bold that we are working in domain and co domain of real numbers so everything he says only applies there so I got used to saying things like x^2=1 has no solutions and all but I guess since here its on general forum i should have specified.