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Chapter 3: Problem 6

Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval. $$ f(x)=2 x+4 ; \quad[-1,1] $$

Short Answer

Expert verified
Minimum value is 2 at \( x = -1 \), maximum value is 6 at \( x = 1 \).

Step by step solution

01

Identify Critical Points

Calculate the derivative of the function to find critical points. The function is given as \( f(x) = 2x + 4 \). The first derivative is \( f'(x) = 2 \). Since \( f'(x) = 2 \) is never zero, there are no critical points.
02

Evaluate the Function at Endpoints

Evaluate the function at the endpoints of the interval \( [-1, 1] \). Compute \( f(-1) \) and \( f(1) \): \( f(-1) = 2(-1) + 4 = 2 \) and \( f(1) = 2(1) + 4 = 6 \).
03

Compare the Values

Compare the function values obtained to determine the absolute maximum and minimum values over the given interval. \( f(-1) = 2 \) is the minimum value and \( f(1) = 6 \) is the maximum value.

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Key Concepts

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

Critical Points
In calculus, critical points are where the function's derivative is zero or undefined. To find these points for any function, you start by finding the first derivative. For instance, with our function \( f(x) = 2x + 4 \), the first derivative is \( f'(x) = 2 \). Notice that this derivative is a constant and does not equal zero anywhere, and it is not undefined.
This means that, for this specific function, there are no critical points. These points are valuable because they're where the function can possibly have a local maximum or minimum. Remember, you need critical points to determine where the slopes of the function equal zero or change abruptly.
Function Evaluation
To find the absolute maximum and minimum values of a function on a specific interval, you evaluate the function at crucial points鈥攖hese include both the critical points and the endpoints of the interval.
In the given problem, our function \( f(x) = 2x + 4 \) must be evaluated at the endpoints because there are no critical points. The provided interval is \( [-1, 1] \). Thus, the endpoints are \( x = -1 \) and \( x = 1 \).
Calculate the function values:
  • For \( x = -1 \), \( f(-1) = 2(-1) + 4 = 2 \)
  • For \( x = 1 \), \( f(1) = 2(1) + 4 = 6 \)
These calculations give you the necessary values to identify the absolute maximum and minimum.
Calculus Problem-Solving
Solving calculus problems often means following a systematic approach. For example, finding an interval's absolute maximum and minimum involves these key steps: find the critical points, evaluate the function at the endpoints, and compare the values obtained.
1. Compute the derivative and identify points where it is zero or undefined (critical points).
2. Evaluate the function at both the critical points and the interval's endpoints.
3. Compare these values to determine which are the maximum and minimum. In the interval \( [-1, 1] \) for our function, we evaluated at endpoints as no critical points were found.
The lowest value, or the minimum, was \( f(-1) = 2 \), and the highest value, or the maximum, was \( f(1) = 6 \).
This clear comparison of values illuminates the problem-solving process in calculus.

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