Problem 40 - Calculus Early Transcendentals

Question

In each part sketch a continuous curve \(y=f(x)\) with the stated properties. (a) \(f(2)=4, f^{\prime}(2)=0, f^{\prime \prime}(x)<0\) for all \(x\) (b) \(f(2)=4, f^{\prime}(2)=0, f^{\prime \prime}(x)>0\) for \(x< 2, \quad f^{\prime \prime}(x)< 0\) for \(x>2\) (c) \(f(2)=4, f^{\prime \prime}(x)>0\) for \(x \neq 2\) and \(\lim _{x \rightarrow 2^{+}} f^{\prime}(x)=-\infty, \lim _{x \rightarrow 2^{-}} f^{\prime}(x)=+\infty\)

Step-by-Step Solution

Answer

(a) Maximum point at (2, 4), curve concave down. (b) Inflection point at (2, 4), concave up left, concave down right. (c) Vertical cusp at (2, 4), concave up elsewhere.

Step 1: Graph Analysis for Part (a)

First, for part (a), we need a graph that satisfies:- The point \((2, 4)\) lies on the curve, i.e., \(f(2) = 4\).- \(f'(2) = 0\), indicating a horizontal tangent line at \(x = 2\).- \(f''(x) < 0\) for all \(x\), meaning that the function is concave down everywhere.To sketch the curve, draw a parabola-like curve with a maximum at \(x = 2\). This implies that the curve is opening downwards, and the tangent at \(x = 2\) is horizontal.

Step 2: Graph Analysis for Part (b)

For part (b), sketch a graph with these properties:- \(f(2) = 4\).- \(f'(2) = 0\).- \(f''(x) > 0\) for \(x < 2\), indicating the graph is concave up when \(x < 2\).- \(f''(x) < 0\) for \(x > 2\), indicating the graph is concave down when \(x > 2\).Visualize a curve that changes from concave up to concave down at \(x = 2\). This typically represents an inflection point at \(x = 2\), where the slope transitions from increasing to decreasing, making a shape similar to an \(S\) curve.

Step 3: Graph Analysis for Part (c)

Finally, part (c) requires the curve to satisfy:- \(f(2) = 4\).- \(f''(x) > 0\) for \(x eq 2\), thus the curve is continuously concave up.- \(\lim_{x \rightarrow 2^{+}} f'(x) = -\infty\), indicating a vertical tangent line going downward as \(x\) approaches 2 from the right.- \(\lim_{x \rightarrow 2^{-}} f'(x) = +\infty\), indicating a vertical tangent line going upward as \(x\) approaches 2 from the left.For such behavior, sketch a function that has an upward swoop from the left and a downward swoop from the right at \(x = 2\), resembling a vertical cusp at that point.

Key Concepts

Continuous Curve

A continuous curve is a function that has no breaks, jumps, or holes across its domain. This means you can trace the curve on a graph without lifting your pencil. In calculus, ensuring continuity is vital for applying different theorems and concepts.
To establish a curve's continuity, check if:

• The function is defined at all points in its domain.
• The limit of the function exists at each point in the domain, including endpoints if they exist.
• The value of the function at any point equals the limit of the function as it approaches that point.

Understanding continuous curves is essential when sketching graphs, such as in exercises where you determine points, tangents, and concavity.

Concavity

Concavity describes how a curve bends or how it concaves either upwards or downwards. It's related to the second derivative of the function, denoted as \( f''(x) \). Knowing the concavity helps in understanding the shape and behavior of the curve over specific intervals.
Key points about concavity:

• If \( f''(x) > 0 \), the curve is concave up, creating an upward bending or U-shape.
• If \( f''(x) < 0 \), the curve is concave down, taking on a downward bending or n-shape.
• Changes in concavity occur at points called inflection points.

These properties help predict and illustrate turning points of a curve when sketching graphs.

Inflection Point

An inflection point is where the curve changes concavity from concave up to concave down, or vice versa. It's a key feature in graph sketching as it indicates a change in the function's rate of change.

• At an inflection point, the second derivative, \( f''(x) \), is typically zero or undefined.
• The function's slope increases and then decreases, or vice versa, highlighting a shift in direction.
• Although the second derivative is involved, it's crucial to test points around the candidate inflection point to confirm the change in concavity.

Identifying inflection points provides insight into complex behaviors of functions, especially when multiple curves are involved.

Vertical Tangent

A vertical tangent occurs when the derivative of a function approaches infinity or negative infinity at a certain point. This indicates that the slope of the tangent line at that point is becoming infinitely steep.
Characteristics of a vertical tangent:

• There's a drastic change in slope as x approaches the point from either one side or both.
• The tangent line is vertical, meaning it has an undeclared slope.
• This is seen in limits where \( \lim_{x \to a^+} f'(x) = +\infty \) or \( \lim_{x \to a^-} f'(x) = -\infty \).

Understanding vertical tangents is important when analyzing the local behavior of graphs, especially near points of discontinuity or sharp turns.