Problem 10 - University Calculus: Early Transcendentals

Question

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=6-2 x-x^{2}$$

Step-by-Step Solution

Answer

The local and absolute maximum is at (-1, 7); no minimum or inflection points exist.

Step 1: Identify the derivative

To find the extreme and inflection points, first determine the derivative of the function. Given \[y = 6 - 2x - x^2,\]we find the first derivative, \[y' = \frac{d}{dx}(6 - 2x - x^2) = -2 - 2x.\]

Step 2: Find critical points

Critical points occur where the derivative is zero or undefined. Set \[y' = -2 - 2x = 0\]and solve for \(x\). \[-2 - 2x = 0 \-2 = 2x \x = -1.\]Thus, there is a potential extreme point at \(x = -1\).

Step 3: Determine extreme point type

Use the second derivative test to classify the critical point. Find the second derivative:\[y'' = \frac{d}{dx}(-2 - 2x) = -2.\]Since \(y'' < 0\), the function is concave down at \(x = -1\), indicating a local maximum.

Step 4: Calculate function value at critical point

Substitute \(x = -1\) into the original function to find the y-coordinate:\[y = 6 - 2(-1) - (-1)^2 = 6 + 2 - 1 = 7.\]The local maximum is \((-1, 7)\).

Step 5: Identify inflection points

Inflection points occur where the second derivative changes sign. Since \[y'' = -2\]is constant and never zero or undefined, there are no inflection points.

Step 6: Determine the absolute extreme

Given that the parabola opens downwards (as indicated by the leading negative coefficient \(-x^2\)), the highest point is the vertex. The local maximum at \((-1, 7)\) is also the absolute maximum over all real numbers. There is no absolute minimum since the parabola extends downwards indefinitely.

Key Concepts

Extreme Points

In calculus, extreme points of a function are key markers that help identify the highest or lowest values of the function within a certain interval. These points can either be local or absolute extreme points.

• Local Extreme Points: These are points where the function takes on a maximum or minimum value relative to nearby points.
• Absolute Extreme Points: These are points where a function achieves its highest or lowest value across its entire domain.

To determine extreme points, we start by finding the first derivative of the function, which is used to locate critical points. These critical points occur where the derivative is zero or does not exist. At these points, a function can have either a peak (maximum) or a trough (minimum).
In our quadratic function, we identified the extreme point at \(x = -1\), making \(-1, 7\) a local maximum, which also serves as the absolute maximum point because the parabola opens downward.

Derivative

Derivatives are foundational in calculus, acting as mathematical tools to describe how a function changes as its input changes.

• The derivative \(f'(x)\) gives us the slope of the tangent line to the function at any point.
• A zero derivative indicates a horizontal tangent, often pointing towards a local extreme point.

To analyze the function \(y = 6 - 2x - x^2\), we compute the first derivative, resulting in \(y' = -2 - 2x\). By setting \(y' = 0\), we find that \(x = -1\) is a critical point. This zero slope signifies a possible extremum, confirming that the rate of change at that point could either be switching from increasing to decreasing, or vice versa.

Second Derivative

The second derivative of a function gives us insight into its curvature or concavity. It tells us about the acceleration or deceleration of the function's slope.

• If \(f''(x) > 0\), the function is concave up, resembling the shape of a "U" and suggesting that a minimum point might be present.
• If \(f''(x) < 0\), the function is concave down, resembling an "n" and potentially indicating a maximum point.

For our function, the second derivative is constant: \(y'' = -2\). Since \(y'' < 0\), the concavity is downward across the domain. This reinforces the idea that the critical point at \(x = -1\) represents a local maximum because the graph bends downward.

Concavity

Understanding concavity is crucial for interpreting the shape of a graph around certain points. It is determined by the sign of the second derivative. Concavity plays a role in predicting the behavior of functions beyond just locating extreme points.

• Concave Up: If the second derivative is positive, the function curves upwards like a cup, which generally indicates the presence of a local minimum.
• Concave Down: When the second derivative is negative, it implies the function curves downward, hinting at a local maximum.

In our example, since the second derivative \(y'' = -2\) is negative, the function \(y = 6 - 2x - x^2\) is concave down throughout its entire domain. This concave shape ensures that the only extreme point \((-1, 7)\) is indeed a maximum, both locally and absolutely, because the entire graph arches downward. Concavity helps us visualize how the function behaves around critical points, as well as informs us of the absence of inflection points since the concavity doesn't change.