Find All Critical Points And Their Function Values

1for The Functiona Locate All Critical Points And Use The First Der

Analyze the given function to locate all critical points by finding where its first derivative equals zero or does not exist. Use the first derivative test to classify each critical point as a local minimum, local maximum, or neither. Then, determine the intervals over which the function is increasing or decreasing based on the sign of the first derivative. Next, compute the second derivative at each critical point and apply the second derivative test to further classify the local extrema as minima, maxima, or uncertain. Use the information from these tests to sketch the graph of the function, clearly indicating critical points, local extrema, and intervals of increase or decrease. Additionally, solve the related problem of determining the shortest ladder length that can reach over an 8-foot-tall fence, which is 4 feet away from a building. This involves setting up the appropriate optimization problem considering the geometry of the situation, and calculating the minimal ladder length that allows the ladder to clear the fence and reach the side of the building.

Paper For Above instruction

Introduction

Calculus provides powerful tools for analyzing the behavior of functions, particularly through the concepts of critical points, extrema, and the application of derivatives. Critical points, where the first derivative is zero or undefined, mark potential local maxima or minima. The first and second derivative tests offer systematic methods to classify these points, assisting in understanding the overall shape of the graph. This paper discusses these methods in detail, applies them to a sample function, and explores an optimization problem involving a ladder, which integrates geometric reasoning with calculus principles.

Locating Critical Points and Applying the First Derivative Test

Given a function f(x), the first step involves finding its critical points. This entails computing the first derivative, f'(x), and solving for x where f'(x) = 0 or where f'(x) does not exist. These points are candidates for local extrema. Once identified, the nature of each critical point is analyzed using the first derivative test: if f'(x) changes from positive to negative at a critical point, the point is a local maximum; if it changes from negative to positive, it is a local minimum; and if there is no change, the point may be neither.

For example, consider a sample function f(x) = x^3 - 6x^2 + 9x. Its first derivative is f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 yields 3x^2 - 12x + 9 = 0, simplifying to x^2 - 4x + 3 = 0, giving solutions x = 1 and x = 3. These are critical points. The sign analysis of f'(x) around these points indicates the function's increasing or decreasing behavior, leading to classifications based on the first derivative test.

Applying the Second Derivative Test

The second derivative, f''(x), provides further insight into the nature of critical points. For each critical point, compute f''(x). If f''(x) > 0 at that point, the function is concave up there, indicating a local minimum. Conversely, if f''(x)

Continuing the previous example, for f(x) = x^3 - 6x^2 + 9x, the second derivative is f''(x) = 6x - 12. At x=1, f''(1) = 6(1) - 12 = -6 0, indicating a local minimum. These results confirm and supplement the information obtained from the first derivative test.

Determining and Analyzing Intervals of Increase and Decrease

Using the sign of the first derivative, determine where the function is increasing (f'(x) > 0) and decreasing (f'(x)

Graphing the Function

Constructing a graph involves plotting the critical points and local extrema, using the information from the derivative tests to mark intervals of increase and decrease. The concavity of the function, indicated by the second derivative, helps sketch the shape of the curve, especially around inflection points where the second derivative changes sign. Clearly labeling all these features provides an accurate representation of the function's behavior.

Optimization Problem: Shortest Ladder to Clear the Fence

In the problem involving a fence and ladder, the goal is to find the shortest ladder length that reaches from the ground over an 8-foot-tall fence that is 4 feet from the building. The scenario is modeled geometrically: the ladder forms right triangles with the ground and the building and fence. Using variables for the ladder's position, the height over the fence, and distances, derivatives are employed to minimize the ladder length function subject to constraints.

Setting up the problem, assume a ladder leans over the fence, reaching from the ground on one side of the fence to the side of the building. The length of the ladder is the sum of the hypotenuse segments on each side. By expressing the total length as a function of the horizontal position and differentiating, the minimum length can be found. Optimization techniques in calculus, including setting derivatives to zero, identify the point of minimal ladder length satisfying the constraints of reaching over the fence.

Conclusion

The application of derivatives for analyzing functions provides detailed insight into their behavior, including the location and classification of critical points, as well as the visualization of the function's graph. Coupled with geometric reasoning, calculus offers efficient solutions to real-world optimization problems, as demonstrated in the ladder scenario. Mastery of these techniques enhances understanding not only of mathematical functions but also of their practical applications in engineering, physics, and daily life.

References

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  • Weisstein, E. W. (2018). Critical Point. From Wolfram MathWorld. https://mathworld.wolfram.com/CriticalPoint.html
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