Problem 18 Prove that \(x^{3}+4 x-3=0\) has... [FREE SOLUTION]

Chapter 2: Problem 18

Prove that \(x^{3}+4 x-3=0\) has exactly one solution.

Short Answer

Expert verified

The equation \(x^{3}+4 x-3=0\) has exactly one solution. This is determined by demonstrating that the function \(x^{3}+4 x-3\) is strictly increasing over the entire domain of real numbers, meaning that it can only cross the x-axis once, indicating one real solution.

Step by step solution

Find the derivative of the function

The first derivative of the function \(x^{3}+4 x-3\) can be calculated by applying the power rule for differentiation, which states that the derivative of \(x^n\) is \(n*x^{n-1}\). By applying this rule, the derivative would be \(3x^{2}+4\).

Analyze the sign of the derivative

The next step involves determining the sign of the derivative. The function \(3x^{2}+4\) is always greater than zero for all real \(x\) because \(x^{2}\) is always non-negative, and so \(3x^{2}\) is also non-negative. Adding 4 to a non-negative number results in a positive number. Thus, the derivative is positive for all real \(x\). This implies that the original function \(x^{3}+4 x-3\) is strictly increasing for all real \(x\).

Determine the number of solutions

Since the function \(x^{3}+4 x-3\) is continuously increasing, it can only cross the x-axis once, signifying that the function has exactly one root or solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative

In calculus, the derivative of a function is a measure of how a function's output value changes as its input changes. For the function, \(f(x) = x^3 + 4x - 3\), finding the derivative involves applying basic differentiation rules. The main rule applied here is the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\).
This allows you to determine the rate at which \(f(x)\) changes for small changes in \(x\).

First, understand the role of derivatives in analyzing functions. It helps identify increasing and decreasing behavior of the function.
Applying the power rule yields the derivative: \(f'(x) = 3x^2 + 4\).

By evaluating this derivative, we gain insights into the nature of the original function's graph.

Strictly Increasing Function

A strictly increasing function means that as the input \(x\) increases, the output \(f(x)\) also increases.
The condition for a function \(f(x)\) to be strictly increasing is that its derivative \(f'(x)\) is positive for all input values.

In our function \(x^3 + 4x - 3\), the derivative \(3x^2 + 4\) is always positive since the square of any real number \(x^2\) is non-negative.
Adding a positive constant like 4 to it ensures that \(f'(x)\) is greater than zero for all \(x\).

Therefore, the function \(x^3 + 4x - 3\) is strictly increasing, confirming it only rises as \(x\) increases. This helps determine how many times it crosses the x-axis.

Power Rule

The Power Rule is an essential tool in calculus that simplifies the process of finding derivatives for polynomial functions. It states that the derivative of \(x^n\), where \(n\) is any real number, is \(nx^{n-1}\).

For example, the derivative of \(x^3\) is \(3x^2\).
The Power Rule is particularly useful because it can be applied independently to each term of a polynomial.

This rule makes it efficient to differentiate functions like \(f(x) = x^3 + 4x - 3\) term-by-term, allowing for quick calculation and analysis of the slope of the function.

Root of a Polynomial

The root of a polynomial is the value of \(x\) where the polynomial equals zero. Finding roots is equivalent to solving for solutions to the equation derived from setting the polynomial to zero.
For the polynomial \(x^3 + 4x - 3\), a root is where the graph of the function intersects the x-axis.

Since the function is strictly increasing, it implies it has exactly one point where it equals zero.
Visualizing this means the function will continually rise, crossing the x-axis exactly once.

This guarantees the equation has one real solution or root.

First Derivative Test

The First Derivative Test is a method used to determine where a function is increasing or decreasing and identify local maxima or minima. When applied to \(f(x) = x^3 + 4x - 3\), since \(f'(x) = 3x^2 + 4\) is always positive:

This indicates the function does not switch from increasing to decreasing or vice versa.
As a result, there are no local maxima or minima, confirming that the function steadily increases without any turning points.

The first derivative test thus reinforces that the graph's behavior is continuously rising, ensuring only one real root.

91影视

Prove that \(x^{3}+4 x-3=0\) has exactly one solution.

Short Answer

Step by step solution

Find the derivative of the function

Analyze the sign of the derivative

Determine the number of solutions

Key Concepts

Derivative

Strictly Increasing Function

Power Rule

Root of a Polynomial

First Derivative Test

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