gb7nash said:I don't think the point of this problem is to find the roots, but just to show they exist. Are you allowed to use the intermediate value theorem?
ehild said:Find out the behaviour of the function as x goes either to infinity and to minus infinity. It is also useful to calculate f(0), f(1), f(-1), so as you get a picture of the function. Does it have minima and maxima? how many?
No, this isn't true. As x approaches -∞, f(x) approaches +∞.frenchkiki said:Yes I am allowed to use the intermediate value theorem.
as tends to inf: lim f(x) = inf
as tends to -inf: lim f(x) = -inf
The problem is not asking for this. Focus on what the problem is asking you to show. It might be helpful to sketch a rough graph of this function based on the two limits and the three function values above. Keep in mind the suggestion from gb7nash.frenchkiki said:f(0)=-1
f(1)=-1
f(-1)=1
I'm not sure on how to find the extremum though
frenchkiki said:as tends to inf: lim f(x) = inf
as tends to -inf: lim f(x) = -inf
f(0)=-1
f(1)=-1
f(-1)=1
I'm not sure on how to find the extremum though
To prove that this equation has two real roots, we can use the Intermediate Value Theorem. This theorem states that if a continuous function f(x) has values of opposite signs at two points a and b, then there must be at least one root between a and b. In this case, we can use f(x) = x^4 - x - 1 and show that it has values of opposite signs at x = -1 and x = 1, proving the existence of at least two real roots.
No, the quadratic formula only applies to equations of the form ax^2 + bx + c = 0. This equation is a quartic equation, meaning it has a degree of 4. To solve this equation, we will need to use other methods such as the method of undetermined coefficients or the Rational Root Theorem.
The Rational Root Theorem states that if a polynomial equation with integer coefficients has a rational root (p/q), then p must be a factor of the constant term and q must be a factor of the leading coefficient. In the case of x^4 - x - 1 = 0, the constant term is -1 and the leading coefficient is 1, so the possible rational roots are ±1, ±1/1, ±1/1. Using synthetic division or plugging in these values, we can show that none of them are actually roots of the equation.
Yes, there are other methods such as using numerical approximations like Newton's method or using the Descartes' Rule of Signs to estimate the number of positive and negative roots. However, these methods may not always give exact solutions and may require more advanced mathematics.
Proving the existence of real roots for this equation is important because it shows that this equation has solutions that are not complex numbers. In real analysis, we often deal with real-valued functions and equations, so knowing the number of real roots can help us better understand the behavior of the function and its graph. Additionally, this proof can also be applied to other similar equations and help us make generalizations about their roots.