Problem 36 - Applied Calculus

Question

Assume \(f\) has a derivative everywhere and has just one critical point, at \(x=3 .\) In parts \((a)-(d)\), you are given additional conditions. In each case decide whether \(x=3\) is a local maximum, a local minimum, or neither. Explain your reasoning. Sketch possible graphs for all four cases. (a) \(f^{\prime}(1)=3\) and \(f^{\prime}(5)=-1\) (b) \(f(x) \rightarrow \infty\) as \(x \rightarrow \infty\) and as \(x \rightarrow-\infty\) (c) \(f(1)=1, f(2)=2, f(4)=4, f(5)=5\) (d) \(f^{\prime}(2)=-1, f(3)=1, f(x) \rightarrow 3\) as \(x \rightarrow \infty\)

Step-by-Step Solution

Answer

(a) Local maximum; (b) Neither; (c) Local minimum; (d) Neither.

Step 1: Analyze Condition (a)

To determine if \(x=3\) is a local maximum or minimum, observe that \(f'(1)=3\), which means the derivative is positive at \(x=1\), and \(f'(5)=-1\), which indicates a negative derivative at \(x=5\). This suggests that \(f'(x)\) changes from positive to negative as \(x\) increases through 3, indicating a local maximum at \(x=3\).

Step 2: Analyze Condition (b)

The condition \(f(x) \rightarrow \infty\) as \(x \rightarrow \pm\infty\) implies that the function increases indefinitely on both sides of \(x=3\). Therefore, \(x=3\) would be neither a local maximum nor a local minimum as \(f(x)\) increases on either end.

Step 3: Analyze Condition (c)

Given points \(f(1)=1\), \(f(2)=2\), \(f(4)=4\), and \(f(5)=5\), the function values increase to the left and right of \(x=3\). There is no decrease around \(x=3\), so \(x=3\) is likely a local minimum, as no other critical points are present.

Step 4: Analyze Condition (d)

At \(x=2\), \(f'(2)=-1\) indicates a decreasing function, and \(f(x)\rightarrow 3\) as \(x\rightarrow \infty\) suggests leveling off to a horizontal asymptote. Since \(f'(x)\) is undefined at \(x=3\) and there's no further trend data, \(x=3\) is neither a maximum nor a minimum.

Key Concepts

Derivative Analysis

Understanding derivative analysis is crucial to finding critical points in a function. A derivative gives us the rate at which a function is changing at any given point. If the derivative of a function, denoted as \(f'(x)\), is zero at a particular point, that point is termed a "critical point." Critical points are potential candidates for local maxima or minima.
To analyze such points, look at the sign changes in the derivative:

• If \(f'(x)\) changes from positive to negative, it's a sign of a local maximum.
• If \(f'(x)\) changes from negative to positive, it indicates a local minimum.
• If there is no sign change, then the critical point may be neither.

By examining first derivatives around \(x = 3\) in conditions \((a)-(d)\), one can determine the behavior of the function at these critical points.

Local Maximum

A local maximum occurs at a point where a function's value is larger than the values at all points immediately nearby. For instance, if a hilltop is the highest point in the surrounding landscape, it's a local maximum.
Consider condition (a): Here, \(f'(1) = 3\) and \(f'(5) = -1\). The function transitions from increasing before \(x = 3\) to decreasing after, marking \(x = 3\) as a local maximum. This is because \(f'(x)\) changes from positive to negative, a tell-tale sign of a peak.To visualize, imagine a curve rising to \(x=3\) and then descending — this peak indicates the local maximum.

Local Minimum

Local minima are similar to local maxima, but the function value is lower than its immediate surrounding points. Picture this as a valley in a mountain range. The condition (c) presents a good example: With function values steadily increasing past \(x = 3\), without any decline, this point acts like a basin, making \(x=3\) a local minimum.A key in identifying a local minimum is when \(f'(x)\) switches from negative to positive as \(x\) crosses a critical point. In simple visual terms, imagine a curve that dips as it approaches \(x = 3\), bottoms out, and then rises indefinitely.

Function Sketching

Function sketching enables us to visualize how a function behaves, especially at critical points and asymptotes. By sketching, you're uncovering the story of the function's graph. Elements like local maxima, minima, and asymptotic behavior become evident.To illustrate:

• For a local maximum at \(x=3\) in condition (a), sketch a peak.
• For a local minimum at \(x=3\) in condition (c), depict a trough or valley.
• Consider asymptotic behavior for conditions like (b) and (d), where the function trends towards infinity or a horizontal line respectively.

Sketching these can provide a clear visual representation that bolsters comprehension, emphasizing key behaviors and critical points of the function.