# Math Help - prove that this equation has 1 real root

1. ## 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

2. Make it of the form (x+a)(bx^2 + cx + d)

Then prove bx^2 + cx + d has no real roots.

3. $f(x)=x^3+8x-19, \ f'(x)=3x^2+8>0, \ \forall x\in\mathbf{R}$
So, f is strictly increasing.
$\lim_{x\to\infty}f(x)=\infty, \ \lim_{x\to\ -\infty}f(x)=-\infty$ 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.

4. 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