Proving Rolle's Theorem - Finding 2 Real Roots

[roam]

Use Rolle's theorem to show that $$g(x) = 3x^4 +4x^3 +6x^2 -3$$ has exactly two real roots.



3. The Attempt at a Solution :

Rolle's theorem states that if g is continious on the closed interval [a,b] and differentiable on the open interval (a,b). Assuming that g(a) = g(b). Then g'(x₀) = 0 for at least one point x₀ in (a,b).

Looking at the derivative of that function:
$$g'(x) = 12x^3 + 12x^2 + 12x$$

the function is equal to zero when;
$$12x^3 + 12x^2 + 12x = 0$$
$$x (x^2 + x + 1) = 0$$

I used the quadratic formula and I believe x^2 + x + 1 has no real roots which means x = 0 is the only real root for the derivative function. Thus x = 0 is the only "stationary point" on the curve for the original function g(x).

I only found one stationary point! How does this show that "there can be at most 2 real roots"?

How can I use Rolle's theorem to prove that g has at most two real roots?

(I believe at x = 0 the function is negative and it perhaps crosses the x-axis but I don't know how to find the coordinates of the roots!)

 

[tiny-tim]

higher-order version?

Use Rolle's theorem to show that $$g(x) = 3x^4 +4x^3 +6x^2 -3$$ has exactly two real roots.

Hi roam!

the only thing I can think of is that perhaps they mean the higher-order version of Rolle's theorem …

If g(a) = g(b) = g(c) with a < b < c, then there must be an x with a < x < c and g''(x) = 0.

 

[HallsofIvy]

Use Rolle's theorem to show that $$g(x) = 3x^4 +4x^3 +6x^2 -3$$ has exactly two real roots.



3. The Attempt at a Solution :

Rolle's theorem states that if g is continious on the closed interval [a,b] and differentiable on the open interval (a,b). Assuming that g(a) = g(b). Then g'(x₀) = 0 for at least one point x₀ in (a,b).

Looking at the derivative of that function:
$$g'(x) = 12x^3 + 12x^2 + 12x$$

the function is equal to zero when;
$$12x^3 + 12x^2 + 12x = 0$$
$$x (x^2 + x + 1) = 0$$

I used the quadratic formula and I believe x^2 + x + 1 has no real roots which means x = 0 is the only real root for the derivative function. Thus x = 0 is the only "stationary point" on the curve for the original function g(x).

I only found one stationary point! How does this show that "there can be at most 2 real roots"?

How can I use Rolle's theorem to prove that g has at most two real roots?

(I believe at x = 0 the function is negative and it perhaps crosses the x-axis but I don't know how to find the coordinates of the roots!)

I don't know if you have really "used" Rolles theorem but it is true that y'= 0 only at x= 0. Now, look at x= 1. y'(1)= 1(1²+ 1+ 1)= 3> 0. Since y'= 0 only at 0 (and y' is continuous) that tells that y'> 0 and y is increasing for all x> 0. Look at x= -1. y'(-1)= (-1)((-1)²+ (-1)+ 1)= (-1)(1)= -1< 0. That tells you that y'< 0 and y is decreasing for all x< 0. That should be enough to tell you that y cannot have more than rwo zeros.