Problem 35 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\). $$ f(x)=x^{3}-9 x^{2}+27 x-27 $$

Step-by-Step Solution

Answer

The solution to the equation \(f''(x)=0\) is \(x=3\).

Step 1: Find the First Derivative

The first derivative of the function is given by: \(f'(x) = 3x^2-18x+27\).

Step 2: Find the Second Derivative

The second derivative is the derivative of the first derivative, which gives: \(f''(x) = 6x-18\).

Step 3: Solve the Equation \(f''(x)=0\)

Setting the second derivative equal to zero gives the equation \(6x-18=0\). Solving for \(x\), we get \(x=3\).

Key Concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It has two main branches: differential calculus and integral calculus. Differential calculus deals with the rate at which quantities change. Integral calculus, on the other hand, concerns the accumulation of quantities, such as areas under or between curves. These branches are related to each other by the fundamental theorem of calculus. In practice, calculus can solve problems involving motion, area, volume, and rates of change, and it's widely used in fields like physics, engineering, economics, statistics, and more.

When studying calculus, one typically begins by understanding the concept of a derivative, which measures how a function changes as its input changes. The process of finding a derivative is called differentiation. After grasping derivatives, students move on to study integrals and integration, which is essentially the reverse process of differentiation. The example of finding the second derivative of a cubic polynomial function in the exercise is a direct application of differential calculus.

Derivative of a Function

The derivative of a function represents the rate at which the function's value changes with respect to a change in its input variable. This concept lies at the heart of differential calculus. Formally, the derivative of a function at a certain point is the limit of the average rate of change of the function as the interval considered shrinks to zero. When a function, such as a polynomial, is differentiated, it produces another function that gives the slope of the tangent line to the original function's graph at any point.

For example, if you have a function like the cubic polynomial in our exercise, the first derivative is found by applying rules of differentiation. These include the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\), and the constant rule, which states that the derivative of a constant is zero. The solution process involves applying these rules systematically to each term in the function to obtain the first derivative, and then again to find the second derivative.

Critical Points

In the realm of calculus, critical points are places on the graph of a function where the derivative is either zero or does not exist. These points are critical because they can indicate where a function has a local maximum or minimum or a point of inflection. For smooth functions, these are points where the tangent line is horizontal.

Finding critical points is an essential application of derivatives. To determine them, we first find the function's derivative. Then, we set the derivative equal to zero and solve for the variable to find potential critical points. The second derivative can also provide insight: if it is positive at a critical point, we usually have a local minimum; if it's negative, a local maximum. Critical points that result in a second derivative of zero need further analysis to classify. The exercise provided involved finding only the point at which the second derivative equals zero, often associated with points of inflection, where the concavity of the function graph changes.