Question

Prove that the equation \(x^{3}+c x=d\) always has one positive solution and no negative ones.

Short answer

Question: Explain the steps and results in showing that the equation \(x^3 + cx = d\) has exactly one positive solution and no negative solutions. Answer: To show that the equation \(x^3 + cx = d\) has exactly one positive solution and no negative solutions, we followed these steps: 1. Investigate properties of the function: We found the function \(f(x) = x^3 + cx - d\) and its derivative, \(f'(x) = 3x^2 + c\). We concluded that \(f(x)\) is strictly increasing as the derivative is always non-negative and strictly positive for any \(x \neq 0\). 2. Apply the Intermediate Value Theorem (IVT): Using the IVT, we showed that since \(f(0) = -d < 0\) and \(f(\infty) = \infty\), there must exist at least one positive solution \(x\) for the equation \(x^3 + cx = d\). 3. Show that there are no negative solutions: We analyzed the behavior of the function for negative \(x\) values and found that for all negative \(x\), \(f(x) < 0\), meaning there can be no negative solutions. By following these steps, we demonstrated that the equation \(x^3 + cx = d\) has exactly one positive solution and no negative solutions.

Step-by-step solution

Investigate properties of the function

The given equation can be written as a function: $$ f(x) = x^3 + cx - d $$ To analyze the properties of this function, we can find its derivative. The derivative of the function is: $$ f'(x) = 3x^2 + c $$ Since both \(x^2\) and \(c\) are non-negative, we can see that the derivative is always non-negative, meaning \(f(x)\) is a non-decreasing function. Additionally, since \(x^2\) is strictly positive for any \(x \neq 0\), we can say that \(f(x)\) is strictly increasing.

Apply the Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function takes values \(u\) and \(v\) at points \(a\) and \(b\), then it takes any value between \(u\) and \(v\) at some point between \(a\) and \(b\). In other words, if \(f(a) < u < f(b)\), there exists at least one value \(c\) within the interval \([a, b]\) such that \(f(c) = u\). Taking the endpoints as \(x = 0\) and \(x \rightarrow \infty\), we see that: $$ f(0) = -d < 0 \\ f(\infty) = \infty $$ Therefore, by the Intermediate Value Theorem, there must exist at least one positive solution \(x\) such that \(f(x) = 0\), or \(x^3 + cx = d\).

Show that there are no negative solutions

Since the function \(f(x)\) is strictly increasing, it can only cross the \(y\)-axis once. We've shown that there exists a positive solution, so we only need to confirm that it isn't possible for there to be any negative solutions as well. To do this, we analyze the behavior of the function for negative \(x\) values: $$ x < 0 \Rightarrow x^3 < 0 \\ cx \Rightarrow c (x < 0) < 0 \\ x^3 + cx < 0 $$ Thus, for all negative values of \(x\), \(f(x) < 0\), meaning that there can be no negative solutions. From these three steps, we have demonstrated that the equation \(x^3 + cx = d\) has exactly one positive solution and no negative solutions.

Key concepts

Intermediate Value Theorem

The Intermediate Value Theorem is a fundamental concept in calculus, essential for understanding the behavior of continuous functions. This theorem assures us that if we have a continuous function that takes different values at two points, it must take every intermediate value between these at some point within that interval. Consider a function \( f(x) \) defined on an interval \([a, b]\). If \( f(a) < n < f(b) \), the theorem guarantees there exists a point \( c \) in the interval \( [a, b] \) such that \( f(c) = n \).

In our exercise, this theorem is applied to a cubic function \( f(x) = x^3 + cx - d \), which is continuous on the real number line. By evaluating the function at the endpoints: \( f(0) = -d \) (which is negative) and \( f(x) = \infty \) as \( x \rightarrow \infty \), we observe a change of sign. This change implies there must exist a positive value of \( x \) where \( f(x) = 0 \).

This condition ensures the existence of at least one positive root, confirming that it's impossible for the cubic equation \(x^3 + cx = d\) to lack a solution.

Derivative Analysis

Analyzing derivatives is crucial to understanding the behavior of functions, specifically how they increase or decrease. The derivative of a function provides instantaneous rate of change. For the function in the exercise, \( f(x) = x^3 + cx - d \), we calculate its derivative as \( f'(x) = 3x^2 + c \).

Here, the derivative \( f'(x) = 3x^2 + c \) is always non-negative for all real numbers, because both \( 3x^2 \) and \( c \) are non-negative due to the nature of squares and non-negative constants. This derivative characteristic tells us that the function is non-decreasing. Moreover, because \( 3x^2 \) is strictly positive for any \( x eq 0 \), \( f(x) \) is strictly increasing.

This strictly increasing property is particularly important because it guarantees that \( f(x) \) can only cross the \( x \)-axis once. This means that the sole root identified by the Intermediate Value Theorem must be the only real root of the equation, affirming the existence of one positive solution and excluding any negative solutions.

Cubic Equation Solution

Solving cubic equations can sometimes be intricate, but certain properties can simplify the process. A cubic equation like \( x^3 + cx = d \) structured as a function \( f(x) = x^3 + cx - d \) can offer insights into its roots when analyzed properly.

In this specific case, our previous discussions around the Intermediate Value Theorem and derivative analysis revealed vital characteristics. The Intermediate Value Theorem ensures a positive root by the change of sign across \([0, \infty)\). Meanwhile, the strictly increasing nature as established through derivative analysis negates the presence of multiple roots across any interval, ensuring no negative roots.

Thus, solving this cubic equation focuses on understanding these mathematical concepts rather than computing the roots directly. Recognizing the non-decreasing, in fact strictly increasing behavior, enables us to affirm that the equation \( x^3 + cx = d \) indeed has exactly one positive solution.