# Number of solutions

• November 13th 2008, 07:43 AM
Number of solutions
Prove that $f(x) = 4x^4+2x^2-6x+2=0$ has exactly two solutions.

Proof so far.

Now $f'(x)=4x^3+4x-6$, set it equals to zero, we have $2x^3+2x-3=0$, so this equation have 3 solutions, meaning we have 3 extremas, meaning the equation can cross the x-axis twice.

Would that be enough to prove it?
• November 13th 2008, 08:21 AM
Arch_Stanton
$f'(x)=[4x^3+4x-6]'=12x^2+4$
$\forall \mbox{ }12x^2+4>0$
The derivative is always positive, so value of this finction constantly rises and there's just one root (solution).
Is that what you meant?
• November 13th 2008, 08:34 AM
HallsofIvy
Quote:

Prove that $f(x) = 4x^3+4x-6=0$ has exactly two solutions.

Proof so far.

Now $f'(x)=4x^3+4x-6$

No, that is NOT f', it is just f itelf!

Quote:

, set it equals to zero, we have $2x^3+2x-3=0$, so this equation have 3 solutions, meaning we have 3 extremas, meaning the equation can cross the x-axis twice.

Would that be enough to prove it?
No, [tex]2x^3+ 2x- 3= 0[/itex] is just your original equation divided by 2. IF it has 3 solutions, then the original certainly does NOT have exactly two solutions! The equation you start with has only one real solution and two complex solutions. Did you miswrite the function?
• November 13th 2008, 08:36 AM
Oh, it should have been $x^4+2x^2-6x+2$, I accidentally put in the derivative instead of the original function.