Question

f(x) = x^3 + 8x - 19
Show that the equation f(x)=0 has only one real root.

Answer 1

I can only recommend the Newton-Method or the Method of Bisection for this problem.

Answer 2

It is a polynomial of the 3rd degree.
Look at the end-behaviour of the function:
The limit of f(x) if x => infinity is also infinity.
The limit of f(x) if x => -infinity is also -infinity.
f is continuous for every x, so it must be 0 at least one time.
Now look at the derivative:

f'(x)=3x²+8.

What do you know about the possible values of f' ?

Answer 3

Yeah, you know it has to have one because there is a change in sign basically. And are you saying that f'(x) will never be zero so there's no turning point?

Answer 4

Or that it's always positive.

Answer 5

That is just what I wanted to say ;)
So f is always increasing...

Answer 6

Good man!

Answer 7

YW!