# prove that this equation has 1 real root

• Jun 15th 2008, 07:14 AM
Bryn
prove that this equation has 1 real root
y=x^3+8x-19

I know the derivative is 3x^2+8, but how does this tell me there is no turning point and therefore that the function is increasing has one root

thanks

Bryn
• Jun 15th 2008, 07:19 AM
sean.1986
Make it of the form (x+a)(bx^2 + cx + d)

Then prove bx^2 + cx + d has no real roots.
• Jun 15th 2008, 07:25 AM
red_dog
$\displaystyle f(x)=x^3+8x-19, \ f'(x)=3x^2+8>0, \ \forall x\in\mathbf{R}$
So, f is strictly increasing.
$\displaystyle \lim_{x\to\infty}f(x)=\infty, \ \lim_{x\to\ -\infty}f(x)=-\infty$ and f is continuous, so f has at least a real root. But f is injective (because is strictly increasing) and the root is unique.
• Jun 15th 2008, 08:07 AM
CaptainBlack
Quote:

Originally Posted by Bryn
y=x^3+8x-19

I know the derivative is 3x^2+8, but how does this tell me there is no turning point and therefore that the function is increasing has one root

thanks

Bryn

See this thread. There is an entirly different proof there from any in this thread as yet.

RonL