Number Of Real Roots Of A Function, Given One Of Its Derivatives Has Non Real Roots.

Introduction

In calculus, the study of roots and derivatives is a crucial aspect of understanding the behavior of functions. A root of a function is a value of the input variable that makes the function equal to zero. Derivatives, on the other hand, are used to study the rate of change of a function with respect to its input variable. In this article, we will explore the relationship between the number of real roots of a function and its derivative, specifically when the derivative has non-real roots.

The Relationship Between Roots and Derivatives

Let $f(x)$ be a $n^{th}$ degree polynomial with '$r$' real roots. This means that $f(x)$ can be written in the form:

$$f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$$

where $a_n \neq 0$ and $a_0 \neq 0$. The derivative of $f(x)$ is denoted by $g(x)$ and is given by:

$$g(x) = f'(x) = n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \ldots + a_1$$

Now, suppose that $g(x)$ has non-real roots. This means that there exists a complex number $z$ such that $g(z) = 0$. We can use this information to draw conclusions about the number of real roots of $f(x)$.

The Statement

If $g(x)$ is a derivative of $f(x)$ and $g(x)$ has not all real roots, then it follows that $f(x)$ will not have all real roots.

Proof

To prove this statement, we can use the following argument:

• Suppose that $f(x)$ has all real roots. This means that $f(x)$ can be written in the form:

$$f(x) = a_n (x - r_1) (x - r_2) \ldots (x - r_n)$$

where $r_1, r_2, \ldots, r_n$ are the real roots of $f(x)$.

• Taking the derivative of $f(x)$, we get:

$$g(x) = f'(x) = a_n \left[ (x - r_2) \ldots (x - r_n) + (x - r_1) (x - r_3) \ldots (x - r_n) + \ldots + (x - r_1) \ldots (x - r_{n-1}) \right]$$

• Since $g(x)$ has non-real roots, there exists a complex number $z$ such that $g(z) = 0$. This means that:

$$a_n \left[ (z - r_2) \ldots (z - r_n) + (z - r_1) (z - r_3) \ldots (z - r_n) + \ldots + (z - r_1) \ldots (z - r_{n-1}) \right] = 0$$

• Since $a_n \neq 0$, we can divide both sides of the equation by $a_n$ to get:

$$(z - r_2) \ldots (z - r_n) + (z - r_1) (z - r_3) \ldots (z - r_n) + \ldots + (z - r_1) \ldots (z - r_{n-1}) = 0$$

• This equation can be rewritten as:

$$(z - r_1) \left[ (z - r_2) \ldots (z - r_n) + (z - r_3) \ldots (z - r_n) + \ldots + (z - r_{n-1}) \right] = 0$$

• Since $z$ is a complex number, we know that $z - r_1 \neq 0$. Therefore, we can divide both sides of the equation by $z - r_1$ to get:

$$(z - r_2) \ldots (z - r_n) + (z - r_3) \ldots (z - r_n) + \ldots + (z - r_{n-1}) = 0$$

• This equation can be rewritten as:

$$(z - r_2) \left[ (z - r_3) \ldots (z - r_n) + \ldots + (z - r_{n-1}) \right] = 0$$

• Since $z$ is a complex number, we know that $z - r_2 \neq 0$. Therefore, we can divide both sides of the equation by $z - r_2$ to get:

$$(z - r_3) \ldots (z - r_n) + \ldots + (z - r_{n-1}) = 0$$

• Continuing this process, we can show that:

$$(z - r_n) = 0$$

• This means that $z = r_n$, which is a real number. Therefore, we have shown that if $f(x)$ has all real roots, then $g(x)$ must also have all real roots.

Conclusion

In this article, we have explored the relationship between the number of real roots of a function and its derivative, specifically when the derivative has non-real roots. We have shown that if $g(x)$ is a derivative of $f(x)$ and $g(x)$ has not all real roots, then it follows that $f(x)$ will not have all real roots. This result has important implications for the study of roots and derivatives in calculus.

Examples

To illustrate this result, let's consider a few examples:

• Suppose that $f(x) = x^3 - 6x^2 + 11x - 6$. This function has three real roots: $x = 1, 2, 3$. The derivative of $f(x)$ is given by:

$$g(x) = f'(x) = 3x^2 - 12x + 11$$

• The roots of $g(x)$ are given by:

$$3x^2 - 12x + 11 = 0$$

• Solving this equation, we get:

$$x = \frac{12 \pm \sqrt{144 - 132}}{6} = \frac{12 \pm \sqrt{12}}{6}$$

• Therefore, the roots of $g(x)$ are complex numbers. This means that $f(x)$ has not all real roots, even though it has three real roots.

• Suppose that $f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1$. This function has four real roots: $x = 1, 1, 1, 1$. The derivative of $f(x)$ is given by:

$$g(x) = f'(x) = 4x^3 - 12x^2 + 12x - 4$$

• The roots of $g(x)$ are given by:

$$4x^3 - 12x^2 + 12x - 4 = 0$$

• Solving this equation, we get:

$$x = \frac{12 \pm \sqrt{144 - 144}}{12} = \frac{12}{12} = 1$$

• Therefore, the roots of $g(x)$ are real numbers. This means that $f(x)$ has all real roots, even though its derivative has a repeated root.

Applications

The result we have proved has important implications for the study of roots and derivatives in calculus. For example:

• Suppose that we want to find the number of real roots of a function $f(x)$ that has a certain number of real roots. We can use the result we have proved to show that if the derivative of $f(x)$ has non-real roots, then $f(x)$ will not have all real roots.
• Suppose that we want to find the number of real roots of a function $f(x)$ that has a certain number of complex roots. We can use the result we have proved to show that if the derivative of $f(x)$ has real roots, then $f(x)$ will have a certain number of real roots.

Future Work

There are many open questions in the study of roots and derivatives in calculus. For example:

• Suppose that we want to find the number of real roots of a function $f(x)$ that has a certain number of real roots and a certain number of complex roots. We can use the result we have proved to show that if the derivative of $f(x)$ has non-real roots, then $f(x)$ will not have all real roots.
• Suppose that we want to find the number of real roots of a function $f(x)$ that has a certain number of real roots and a certain number of complex roots. We can use the result we have proved to show that if the derivative of $f(x)$ has real roots, then $f(x)$ will have a certain number of real roots.

Conclusion

Q: What is the relationship between the number of real roots of a function and its derivative?

A: The relationship between the number of real roots of a function and its derivative is a crucial aspect of understanding the behavior of functions in calculus. If the derivative of a function has non-real roots, then the function will not have all real roots.

Q: Can you provide an example to illustrate this result?

A: Suppose that $f(x) = x^3 - 6x^2 + 11x - 6$. This function has three real roots: $x = 1, 2, 3$. The derivative of $f(x)$ is given by:

$$g(x) = f'(x) = 3x^2 - 12x + 11$$

• The roots of $g(x)$ are given by:

$$3x^2 - 12x + 11 = 0$$

• Solving this equation, we get:

$$x = \frac{12 \pm \sqrt{144 - 132}}{6} = \frac{12 \pm \sqrt{12}}{6}$$

• Therefore, the roots of $g(x)$ are complex numbers. This means that $f(x)$ has not all real roots, even though it has three real roots.

Q: What are some applications of this result?

A: The result we have proved has important implications for the study of roots and derivatives in calculus. For example:

• Suppose that we want to find the number of real roots of a function $f(x)$ that has a certain number of real roots. We can use the result we have proved to show that if the derivative of $f(x)$ has non-real roots, then $f(x)$ will not have all real roots.
• Suppose that we want to find the number of real roots of a function $f(x)$ that has a certain number of complex roots. We can use the result we have proved to show that if the derivative of $f(x)$ has real roots, then $f(x)$ will have a certain number of real roots.

Q: Can you provide some examples of functions that have non-real roots?

A: Yes, here are a few examples of functions that have non-real roots:

• $f(x) = x^2 + 1$
• $f(x) = x^3 + 2x^2 + 3x + 4$
• $f(x) = x^4 + 4x^3 + 6x^2 + 4x + 1$

Q: Can you provide some examples of functions that have real roots?

A: Yes, here are a few examples of functions that have real roots:

• $f(x) = x^2 - 4$
• $f(x) = x^3 - 6x^2 + 11x - 6$
• $f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1$

Q: What are some open questions in the study of roots and derivatives in calculus?

A: There are many open questions in the study of roots and derivatives in calculus. For example:

• Suppose that we want to find the number of real roots of a function $f(x)$ that has a certain number of real roots and a certain number of complex roots. We can use the result we have proved to show that if the derivative of $f(x)$ has non-real roots, then $f(x)$ will not have all real roots.
• Suppose that we want to find the number of real roots of a function $f(x)$ that has a certain number of real roots and a certain number of complex roots. We can use the result we have proved to show that if the derivative of $f(x)$ has real roots, then $f(x)$ will have a certain number of real roots.

Q: What are some future directions for research in the study of roots and derivatives in calculus?

A: There are many future directions for research in the study of roots and derivatives in calculus. For example:

• We can use the result we have proved to study the behavior of functions with a certain number of real roots and a certain number of complex roots.
• We can use the result we have proved to study the behavior of functions with a certain number of real roots and a certain number of complex roots.
• We can use the result we have proved to study the behavior of functions with a certain number of real roots and a certain number of complex roots.

Conclusion

In this article, we have explored the relationship between the number of real roots of a function and its derivative. We have shown that if the derivative of a function has non-real roots, then the function will not have all real roots. We have also provided examples to illustrate this result and discussed some applications and open questions in the study of roots and derivatives in calculus.