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Chapter 7: Problem 45

If a function has a critical point at which \(f_{x x}>0\) what can you conclude?

Short Answer

Expert verified
The critical point is a local minimum.

Step by step solution

01

Identify the Critical Point Condition

A critical point occurs where the first derivative of the function, \(f'(x)\), equals zero. This means that the function is at a maximum, minimum, or a saddle point.
02

Apply the Second Derivative Test

The second derivative test helps us determine the nature of the critical point. If \(f''(x) > 0\), this indicates that the function is concave up at that point.
03

Conclusion from Concavity

Since \(f''(x) > 0\) implies the function is concave up, we can conclude that the critical point is a local minimum. In a local minimum, the function reaches a lower value than at surrounding points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Second Derivative Test
The second derivative test is a handy tool in calculus for determining the nature of critical points on a function. Critical points are where a function's first derivative, \( f'(x) \), equals zero or is undefined. These points can represent potential maximums, minimums, or even saddle points. To apply the second derivative test, we need to look at the second derivative, \( f''(x) \).
  • If \( f''(x) > 0 \), the function is concave up at that point, suggesting a local minimum.
  • If \( f''(x) < 0 \), the function is concave down, suggesting a local maximum.
  • If \( f''(x) = 0 \), the test is inconclusive, and further analysis is required.
This test allows us to quickly identify the behavior of the function around critical points, helping us understand the function's shape and potential turning points.
Concave Up Function
A function that is concave up is visually shaped like an upwards-opening bowl or a smile. In more technical terms, a function is concave up on an interval if the graph of the function lies above all its tangent lines within that interval. This property is closely linked to the sign of the second derivative.
  • When \( f''(x) > 0 \), it indicates that the rate of change of the derivative is increasing, causing the function to curve upwards.
  • This creates a situation where the slope of the tangent (the first derivative) is increasing.
  • This means the function forms a valley, often suggesting the presence of a local minimum.
Understanding concavity helps inform decisions about optimization and provides insight into the behavior of complex systems.
Local Minimum
A local minimum of a function is a point where the function's value is lower than all nearby values. However, it may not be the lowest value overall on the function's entire domain. Local minima are crucial in optimization problems and occur frequently in real-world applications, from economics to engineering.
  • To find a local minimum, one often starts by identifying critical points, where \( f'(x) = 0 \).
  • The second derivative test is then applied. If \( f''(x) > 0 \) at a critical point, this confirms a local minimum due to the function's concave up nature.
  • In practical applications, local minima help to identify optimal solutions within restricted parameters.
Recognizing and understanding local minima allow us to make informed predictions and decisions based on the function's behavior in specific intervals.

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Most popular questions from this chapter

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