Problem 36 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\). $$ f(x)=3 x^{3}-9 x+1 $$

Step-by-Step Solution

Answer

The solution to the equation \(f''(x)=0\) for the function \(f(x)=3 x^{3}-9 x+1\) is \(x = 0\).

Step 1: Find First Derivative

Firstly need to compute the first derivative of the function \(f(x)\). Since \(f(x) = 3x^3 - 9x + 1\), apply the power rule for differentiation \(nx^{n-1}\), the first derivative \(f'(x)\) of \(f(x)\) will be \(f'(x) = 9x^2 - 9\).

Step 2: Find Second Derivative

Now compute the second derivative, which is the derivative of the first derivative \(f'(x)\). Take the derivative of \(f'(x) = 9x^2 - 9\). Using the power rule, the second derivative \(f''(x)\) will become \(f''(x) = 18x\).

Step 3: Solve Equation

The final task is to solve the equation \(f''(x) = 0\). Set \(f''(x) = 18x = 0\) and solve for \(x\). This will result in \(x = 0\).

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with change and motion. It introduces us to concepts like derivatives and integrals. These tools help us understand how functions behave and change. Calculus is split mainly into two branches: Differential Calculus and Integral Calculus. Differential Calculus focuses on finding how things change, while Integral Calculus deals with accumulation of quantities.

• Derivatives: They are the building blocks of Differential Calculus. A derivative represents the rate of change of a function with respect to a variable.
• Integrals: These represent total accumulation or the area under the curve of a graph.

Calculus helps in understanding the behavior of equations and enables us to tackle real-world problems involving rates of change and areas under curves.

Differentiation

Differentiation is the process of finding the derivative of a function. Think of it as the mathematical way to determine how a function's value changes as its input changes. It's like checking the speed at which something is happening. For example, the derivative of position with respect to time is velocity.

• The power rule is a basic technique in differentiation. For any function of the form \( x^n \), the derivative is \( nx^{n-1} \).
• In the given exercise, you take the derivative of \( 3x^3 - 9x + 1 \) using the power rule, resulting in \( f'(x) = 9x^2 - 9 \).

Once you have the first derivative, obtaining the second derivative involves differentiating the first derivative again. This tells us about the curvature of the function, indicating how the rate of change itself changes. It's like finding out how velocity changes, which is acceleration.

Finding Roots

Finding roots of an equation is essential in calculus for determining when a certain condition is met. A root is a solution where the equation equals zero. In our context, we're interested in finding when the second derivative, \( f''(x) \), equals zero.

• The second derivative is used to check concavity or the nature of inflection points of the function.
• In the exercise, after finding the second derivative, \( f''(x) = 18x \), we set it equal to zero for solving: \( 18x = 0 \).
• By isolating \( x \), we find that the root is \( x = 0 \).

The root of the second derivative can hint at important aspects of graphs, such as points of inflection, where the graph changes its concavity from concave up to concave down or vice versa. This provides insight into the overall behavior and shape of the function.