Can we find at most two real roots for the equation x^4 + 4x + c = 0?

In summary, we are given an equation and asked to find at most two real roots. We can use the basic principle of graphing to determine that a graph can only change direction at a critical point. By taking the derivative and using the given theorem, we can show that there cannot be two distinct roots for the equation. Therefore, there can only be one real root at most.
  • #1
helpm3pl3ase
79
0
This theorem is confusing me even though it is sittin right in front of me.. I am given an equation x^4 + 4x + c = 0 and asked to find at most two real roots??

I know we need to take the derivative, but from there I am lost.
 
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  • #2
If [tex] f(x) [/tex] is differentiable in the open interval [tex] (a,b) [/tex] and continuous on the closed interval [tex] [a,b] [/tex], then there is at least one point [tex] c [/tex] in [tex] (a,b) [/tex] such that:

[tex] f'(c) = \frac{f(b)-f(a)}{b-a} [/tex]

Assume that there are two real roots [tex] c_{1} [/tex] and [tex] c_{2} [/tex] where [tex] c_{1} < c_{2} [/tex].Then [tex] f(c_{1}) = 0 = f(c_{2}) [/tex].

Thus [tex] 4x^{3} + 4 = 0 [/tex]
 
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  • #3
basic principle of garphing: a graph can only change direction at a critical pont, and not always then.
 
  • #4
x^(4) + 4x + c = 0
The function is a polynomial and is differentiable and continuous. Suppose a and b are distinct roots. There exists a c in which a<c<b such that 0 = f(b) - f(a). Since f'(x)= 4x^(2) + 4>0, f(a) != f(b). This is a contradiction; hence, a and b cannot both be roots.
 

What is the Mean Value Theorem?

The Mean Value Theorem is a fundamental theorem in calculus that states that for a continuous and differentiable function on a closed interval, there exists at least one point where the slope of the tangent line is equal to the average rate of change of the function over that interval.

What is the significance of the Mean Value Theorem?

The Mean Value Theorem is significant because it is used to prove many other important theorems in calculus, such as the First and Second Derivative Tests for local extrema. It also has practical applications in fields such as physics and engineering.

How is the Mean Value Theorem used to find the derivative of a function?

The Mean Value Theorem is used in the proof of the Fundamental Theorem of Calculus, which states that the derivative of a function can be found by evaluating the function at a specific point and taking the limit as the interval approaches that point. This allows us to find the derivative of a function without having to use the limit definition.

What are the conditions for the Mean Value Theorem to hold?

The Mean Value Theorem holds for a continuous and differentiable function on a closed interval. Additionally, the function must be continuous on the closed interval and differentiable on the open interval within that closed interval.

Can the Mean Value Theorem be extended to higher dimensions?

Yes, the Mean Value Theorem can be extended to higher dimensions through the Multivariable Mean Value Theorem. This theorem states that if a scalar-valued function is continuous on a closed and bounded region in the plane, and differentiable on the interior of that region, then there exists at least one point where the gradient of the function is equal to the average rate of change of the function over the region.

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