Problem 25 - Calculus Early Transcendentals

Question

a. Find the critical points of the following functions on the domain or on the given interval. b. Use a graphing utility to determine whether each critical point corresponds to a local maximum, local minimum, or neither. $$f(x)=\frac{x^{3}}{3}-9 x \text { on }[-7,7]$$

Step-by-Step Solution

Answer

Question: Determine the critical points of the function $$f(x) = \frac{x^3}{3} - 9x$$ on the interval $$[-7, 7]$$ and specify whether each critical point is a local maximum, local minimum, or neither. Answer: The critical points of the function are $$x = -3$$ (local minimum) and $$x = 3$$ (local maximum).

Step 1: Find the derivative of the function

To find the critical points, we first need to find the derivative of the given function: $$f(x)=\frac{x^{3}}{3}-9x$$ Using the power rule, the derivative of $$f(x)$$ is: $$f'(x) = \frac{d}{dx}(\frac{x^3}{3} - 9x) = x^{2} - 9$$

Step 2: Find the critical points

Next, we need to find where the derivative is equal to zero or undefined. The derivative is a polynomial, so it is never undefined. We set the derivative equal to zero and solve for $$x$$: $$x^{2} - 9 = 0$$ Factor the equation: $$(x - 3)(x + 3) = 0$$ The critical points are at $$x = -3$$ and $$x = 3$$.

Step 3: Determine the nature of the critical points using a graphing utility

Using a graphing utility, plot the original function, $$f(x) = \frac{x^3}{3} - 9x$$, on the interval $$[-7, 7]$$. Observe the behavior of the function around the critical points. At $$x = -3$$, the slope changes from negative to positive, indicating that the function has a local minimum at this point. At $$x = 3$$, the slope changes from positive to negative, indicating that the function has a local maximum at this point. In conclusion: 1. The critical points of the function $$f(x) = \frac{x^3}{3} - 9x$$ on the interval $$[-7, 7]$$ are $$x = -3$$ and $$x = 3$$. 2. The critical point at $$x = -3$$ is a local minimum. 3. The critical point at $$x = 3$$ is a local maximum.

Key Concepts

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It's like finding the slope of the tangent line to the curve of a function at any given point. In simpler terms, the derivative tells us the rate at which our function is growing or shrinking at any particular point. Finding the derivative involves using rules such as the power rule. For instance, if we have a function like \( f(x) = \frac{x^3}{3} - 9x \), we apply the power rule to each term: - The derivative of \( \frac{x^3}{3} \) is \( x^2 \).- The derivative of \(-9x\) is \(-9\).Thus, the derivative of the function becomes \( f'(x) = x^2 - 9 \). With this new derivative function, we can carry out further analysis to examine the behavior of our original function, by locating critical points where the derivative either equals zero or does not exist.

Local Maximum

A local maximum is a point on a graph where the function reaches a highest value within a certain interval. It's like standing on the top of a hill, where any small step you take leads you downward, no matter the direction. To find a local maximum using calculus, you need to identify critical points and analyze them. Critical points occur where the derivative is zero or undefined. For the function \( f(x) = \frac{x^3}{3} - 9x \), we found the derivative \( f'(x) = x^2 - 9 \). Setting this derivative equal to zero gives us the critical points, \( x = -3 \) and \( x = 3 \). To determine whether a critical point is a local maximum, examine how the derivative changes around it:- If the derivative changes from positive to negative at a critical point, this indicates a local maximum.For \( x = 3 \), the slope changes from positive to negative, confirming it as a local maximum point for the function.

Local Minimum

A local minimum is the opposite of a local maximum, representing the lowest value that a function reaches in a small surrounding region. Imagine a valley where any move you make up the slope takes you higher. To confirm a local minimum, we again examine our critical points. For the given function, \( x = -3 \) and \( x = 3 \) are critical points. The focus here is \( x = -3 \), where the derivative changes from negative to positive, indicating a local minimum. This transition suggests the function is dipping down to its lowest point and then increasing afterward. Checking how the derivative behaves around a critical point is crucial:- If the derivative changes from negative to positive, the critical point is a local minimum.Understanding local minima helps identify the lowest points in a function's plot, providing insight into optimization problems or understanding the function's overall behavior.

Polynomial Function

A polynomial function is an expression involving a sum of powers in one or more variables multiplied by coefficients. These functions have the form \( a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0 \), where each \( a_i \) is a coefficient and \( n \) is a non-negative integer. The given function \( f(x) = \frac{x^3}{3} - 9x \) is a classic example of a polynomial, consisting of terms with powers of \( x \). Polynomial functions are continuous and smooth, meaning they don't have breaks or sharp corners. They can take any form based on the degree, which is determined by the highest power of \( x \):- A linear polynomial like \( 9x \) represents a straight line.- A higher degree polynomial introduces curves, such as cubic polynomials which have an \( x^3 \) term.Studying polynomial functions involves understanding how their graphs behave, including identifying critical points, local maxima, and minima. This understanding is foundational in calculus and real-world applications, like modeling and solving analytical problems.