Problem 45 - Calculus Single Variable

Question

Suppose that \(f\) is continuous on \(\mathbb{R},\) that \(f\) is positive on \((-\infty,-3),(2,4),\) and \((4, \infty),\) and that \(f\) is negative on (-3,2) . If \(F\) is an antiderivative of \(f\) on \(\mathbb{R}\), classify the local extrema of \(F\).

Step-by-Step Solution

Answer

Local maximum at \( x = -3 \); local minimum at \( x = 2 \).

Step 1: Understanding Local Extrema in Terms of Antiderivatives

To classify the local extrema of the antiderivative function \( F \), recall that local extrema occur where the derivative \( f = F' \) changes sign. The points where this sign change occurs indicate where \( F \) could have a local maximum or minimum.

Step 2: Identifying Sign Change Intervals

Given that \( f \) is positive on \((-\infty,-3)\), \((2,4)\), and \((4, \infty)\), and negative on \((-3,2)\), observe the points where \( f \) changes sign: at \( x = -3 \) and \( x = 2 \). These points are potential locations for local extrema of \( F \).

Step 3: Determining Types of Local Extrema

1. At \( x = -3 \), \( f \) shifts from positive to negative, indicating that \( F \) has a local maximum.2. At \( x = 2 \), \( f \) shifts from negative to positive, indicating that \( F \) has a local minimum.

Key Concepts

local extrema

When studying functions, particularly in calculus, local extrema refer to points where a function reaches a local maximum or minimum value compared to its immediate surroundings. For the function to have a local extremum, the derivative (or slope) must change sign at that point. This means a local maximum occurs where the function changes from increasing to decreasing, while a local minimum occurs where it changes from decreasing to increasing.
In the context of the given problem, if a function's derivative changes from positive to negative, it indicates a local maximum because the function stops rising and starts falling. Conversely, if it changes from negative to positive, it reveals a local minimum, marking the point where the function stops falling and starts rising. Identifying these changes in direction is crucial for accurately classifying the local extrema of functions.

continuous function

A continuous function is one that has no breaks, jumps, or holes in its graph. This means that you can draw its curve without lifting your pen from the paper. More formally, at any point within its domain, the limit of the function as it approaches that point should equal the function's value at the point itself.
Continuity is a vital concept in calculus, as many fundamental theorems rely on it, including the Intermediate Value Theorem and Rolle’s Theorem. In our problem, the function \( f \) is given as continuous over the entire real line. This continuous nature ensures that transitions from positive to negative values (or vice versa) for the function \( f \) happen smoothly without abrupt changes or undefined behavior, making it possible to determine where the derivative \( F \) might experience local extrema.

sign change

Sign changes in the context of calculus usually refer to points where a function transitions from positive to negative or from negative to positive. This concept is crucial in finding local extrema of functions' derivatives.
For the function \( f \), identifying these transitions involves looking for intervals where the sign of \( f \) shifts. In the exercise, \( f \) changes sign at \( x = -3 \) and \( x = 2 \). These points are essential to determining where the antiderivative \( F \) possesses local extrema. At \( x = -3 \), the transition from positive to negative suggests a local maximum, while at \( x = 2 \), changing from negative to positive suggests a local minimum.

calculus

Calculus is a branch of mathematics that deals with continuous change. It involves concepts such as derivatives, integrals, limits, and functions. Studying calculus allows us to understand and describe changes in physical quantities.
In this problem, we are focusing on the fundamental concept of an antiderivative, or the reverse of differentiation, which allows us to find the original function given its rate of change. The exercise revolves around the antiderivative \( F \) of the function \( f \), and identifying local extrema based on \( f \'s \) sign changes. These principles embody the core themes of calculus—understanding rates of change and cumulative effects over intervals.