Same qustion, more info. Roots of equations:?

Question

OK, I have to prove that
x^3 -15x +1 = 0
has 3 answers. The domain of the function is [-4, 4]. I think I'm supposed to use the intermediate value theorem.
Any suggestions?

Answer 1

you mean 3 roots
lets see: f(-4) = 61 - 64 = -3
f(-3) = 46 - 27 = 19
f(1)= 2 - 15 = -13
f(4) = 64 - 60 +1 = 5
Since f(-4) has a different sign than f(-3) one root is in [-4,-3]
Since f(-3) has a different sign than f(1) one root is in [-3, 1]
The same for [1,4]
In total there are three real roots.

Answer 2

For a cubic to have exactly 3 roots, you must satisfy the following condition.

Say f(x) = x^3 - 15x + 1

Take x1, x2, x3 and x4 such that: x1
f(x1) * f(x2) <0,
f(x2) * f(x3) <0
and
f(x3) * f(x4) <0

You could also use another function y = 0 and check how many solutions it has with f(x). If the answer is 3, then you have 3 roots.

You could also use numerical methods, which I don't recommend, because it is unnecessarily lengthy.

If you are familiar with calculus, you could also use Rolle's Theorem, where if you have a maximum and a minimum simultaneously in the domain, taking of course, your reference points where f(x) = 0, then your function will have 3 roots.

Good Luck
Ayerhs.

Answer 3

The 3 roots are
x= -3.905895954176237
x= 0.0666864373322236
x= 3.8392095168440132