Problem 5 - Active Calculus

Question

Suppose that \(g\) is a differentiable function and \(g^{\prime}(2)=0 .\) In addition, suppose that on \(1

Step-by-Step Solution

Answer

a. Neither, because g is increasing on both sides of 2. b. \(g''(x)\) is positive near x=2. c. x=2 is an inflection point where the concavity changes.

Step 1: Analyze the given information and conditions

We know that:1. The function g is differentiable.2. The derivative of g at x = 2 is zero, i.e., \(g'(2) = 0\).3. On the interval \(1 < x < 2\), \(g'(x)\) is positive.4. On the interval \(2 < x < 3\), \(g'(x)\) is positive.These pieces of information will help determine the behavior of the function g at x=2.

Step 2: Consider the behavior of g around x=2 for part (a)

Since \(g'(2) = 0\), x = 2 is a critical point.A critical point can be a local maximum, local minimum, or neither.Given that \(g'(x)\) is positive on both sides of x = 2 (i.e., for \(1 < x < 2\) and \(2 < x < 3\)), the function g is increasing on both sides of x = 2.Thus, g cannot have a local maximum or local minimum at x = 2 since there's no change from increasing to decreasing or vice versa. Therefore, it is neither.

Step 3: Describe the behavior of \(g''(x)\) near x=2 for part (b)

Since \(g'(x)\) is positive on either side of x = 2 and \(g'(2) = 0\), we can reason about \(g''(x)\):- If \(g'(x)\) transitions smoothly through zero, then \(g''(2)\) must be positive because the slope of \(g'(x)\) is increasing (illustrating convex behavior).- We can think of g having an inflection point at x = 2 where the concavity changes.

Step 4: Explain what's special about x=2 in part (c)

Besides being a critical number, the value \(x = 2\) is an inflection point since \(g'(x)\) does not change sign and \(g''(x)\) is positive at \(x = 2\). This implies a local concave up behavior around the point.

Key Concepts

Critical Points

A critical point of a function occurs where its derivative is zero or undefined. This means the slope of the tangent line to the function at this point is horizontal. For the function given in the exercise, we know that the derivative of the function, denoted as \(g'(x)\), is zero at \(x=2\). This marks \(x=2\) as a critical point. However, identifying a critical point only informs us about a potential extremum, and further analysis is needed to classify it.

Local Extrema

Local extrema are points where a function reaches a local maximum or minimum. To identify whether a critical point is a local maximum, local minimum, or neither, we need to examine the behavior of the function around this point. In the context of the exercise, since \(g'(x)\) is positive on both sides of \(x=2\), the function \(g(x)\) is increasing before and after \(x=2\). Thus, there is no transition from increasing to decreasing or vice versa. Therefore, \(g(x)\) neither has a local maximum nor a local minimum at \(x=2\). It is simply increasing through this point.

Second Derivative Test

The second derivative test is a method to classify critical points by examining the concavity of the function. If \(\ g''(x) > 0 \) at a critical point, the function is concave up, indicating a local minimum. If \( g''(x) < 0 \) at a critical point, the function is concave down, indicating a local maximum. For the function in the exercise, we know that \(g'(x)\) is positive around \(x=2\), making \(g''(x)\) positive as well. This suggests a concave up behavior at \(x=2\). However, due to the continued increase of the function, \(g(x)\) does not exhibit a local extremum at this point.

Inflection Points

An inflection point occurs where the concavity of the function changes. This is marked by the second derivative changing signs: from positive to negative or vice versa. In this exercise, since \(g'(x)\) is positive and transitions smoothly through zero, \(g(2)\) must be an inflection point rather than a local extreme point. At an inflection point, \(g''(x)\) typically passes through zero. For \(x=2\), the positive second derivative suggests the function \(g(x)\) changes concavity, confirming \(x=2\) as an inflection point.

Concavity

Concavity refers to whether a function curves upwards or downwards. If a function is concave up, it looks like an upward-facing bowl and its second derivative is positive. Conversely, if it is concave down, it resembles a downward-facing bowl with a negative second derivative. For the function \(g(x)\) in the exercise, since \(g''(x)\) is positive near \(x=2\), the function exhibits concavity upwards at this point. The function's increase on both sides with a zero derivative at \(x=2\) implies a smooth upward curve, consistent with a concave up behavior.