Problem 3 - Active Calculus

Question

For each of the functions described below (each continuous on \([a, b])\), state the location of the function's absolute maximum and absolute minimum on the interval \([a, b],\) or say there is not enough information provided to make a conclusion. Assume that any critical numbers mentioned in the problem statement represent all of the critical numbers the function has in \([a, b] .\) In each case, write one sentence to explain your answer. a. \(f^{\prime}(x) \leq 0\) for all \(x\) in \([a, b]\) b. \(g\) has a critical number at \(c\) such that \(a0\) for \(xc\) c. \(h(a)=h(b)\) and \(h^{\prime \prime}(x)<0\) for all \(x\) in \([a, b]\) d. \(p(a)>0, p(b)<0,\) and for the critical number \(c\) such that \(a0\) for \(x>c\)

Step-by-Step Solution

Answer

For part a, the absolute maximum is at \(a\) and the absolute minimum is at \(b\). For part b, the absolute maximum is at \(c\) and no conclusion can be made about the minimum. For part c, the absolute maximum is at \(a\) and \(b\), and the absolute minimum is at some point in between. For part d, the absolute maximum is at \(a\) and the absolute minimum is at \(c\).

Step 1: Part a: Analyzing the behavior of the function

Given: - The derivative of the function, \(f^{\textprime}(x)\), is less than or equal to zero for all \(x\) in \([a, b]\). When the derivative of a function is non-positive, the function is either constant or decreasing over the interval. Thus, the maximum value occurs at the left endpoint, and the minimum occurs at the right endpoint, i.e., the absolute maximum is at \(a\) and the absolute minimum is at \(b\).

Step 2: Part b: Analyzing the critical number

Given: - There is a critical number \(c\) such that \(a 0\) for \(x < c\), and - \(g^{\textprime}(x) < 0\) for \(x > c\). The function is increasing for all \(x\) less than \(c\) and decreasing for all \(x\) greater than \(c\). Hence, \(g(c)\) is a local maximum, and the absolute maximum occurs at \(c\). We need more information about the endpoints to determine the absolute minimum, so no conclusion about the minimum can be made.

Step 3: Part c: Analyzing the concavity of the function

Given: - \(h(a) = h(b)\) and - \(h^{\textprime \textprime}(x) < 0\) for all \(x\) in \([a, b]\). Since \(h^{\textprime \textprime}(x) < 0\), the function is concave down over the interval, meaning it reaches its highest value at one of the endpoints. Given that \(h(a) = h(b)\), exactly two endpoints are equal. Thus, the absolute maximum value occurs at both endpoints \(a\) and \(b\) and is equal to \(h(a)\) or \(h(b)\). The absolute minimum occurs at the point \(d\) in between where the concave down makes a minimum.

Step 4: Part d: Analyzing the critical number and behavior of the function

Given: - \(p(a) > 0\), - \(p(b) < 0\), and - There is a critical number \(c\) such that \(a 0\) for \(x > c\). The function \(p\) is decreasing for all \(x\) less than \(c\) and increasing for all \(x\) greater than \(c\). Thus, there is a local minimum at \(c\). Because of the behavior at the endpoints \(a\) and \(b\) with opposite signs \(\therefore\) there is a higher value at \(a\) and this is the maximum. The absolute maximum occurs at \(a\), and the absolute minimum occurs at \(c\).

Key Concepts

absolute maximum

When we talk about the absolute maximum of a function, we mean the highest value that the function reaches on a specified interval. This highest value can occur at a critical point or at one of the endpoints of the interval. For example, if the function is continuously decreasing on the interval \([a, b]\), the absolute maximum will be at the left endpoint \(a\). In another scenario, if the function increases to a certain point and then decreases, the absolute maximum could be at a critical point within the interval.

absolute minimum

The absolute minimum of a function is the lowest value the function attains on a given interval. Similar to the absolute maximum, this lowest value can occur at a critical point or at one of the endpoints of the interval. For instance, if a function is continually increasing on the interval \([a, b]\), the absolute minimum is at the left endpoint \(a\). Alternatively, if the function decreases to a certain point and then increases, the absolute minimum could be found at that lowest point within the interval.

critical points

Critical points of a function are where the first derivative \( f^{\textprime}(x) \) equals zero or where the derivative does not exist. These points are important because they can indicate local maxima, local minima, or points of inflection. To determine what happens at these points, we often use the first and second derivatives of the function. Critical points within the interval help us find where the function changes behavior, such as moving from increasing to decreasing, which helps in locating absolute maxima or minima.

concavity

The concavity of a function describes how the curve bends. It is determined using the second derivative \( f^{\textprime \textprime}(x) \). If \( f^{\textprime \textprime}(x) < 0 \), the function is concave down, meaning it forms a shape like an upside-down cup and might lead to a local maximum. Conversely, if \( f^{\textprime \textprime}(x) > 0 \), the function is concave up, resembling a cup, which could indicate a local minimum. Knowing the concavity helps us understand the overall shape and behavior of the function over the interval.

derivative analysis

Derivative analysis involves using the first and second derivatives of a function to understand its behavior. The first derivative \( f^{\textprime}(x) \) tells us about the slope of the function - whether it's increasing or decreasing. The second derivative \( f^{\textprime \textprime}(x) \) provides information about the concavity. By analyzing these derivatives, we can identify critical points, determine their nature (local maxima or minima), and understand the broader shape of the function. This approach is essential for solving optimization problems efficiently.