Consider The Following. F(x)= 25x2 Find The Critical Numbers. (Enter Your Answers As A Comma-separated
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Introduction
Understanding the concept of critical numbers is a fundamental aspect of calculus, particularly when analyzing functions for their maximums, minimums, and points of inflection. Critical numbers often serve as potential candidates for local extrema and are essential in the process of optimization problems. In this article, we will explore how to find the critical numbers of the function F(x) = 25x², providing a detailed, step-by-step approach to ensure clarity and comprehension.
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What Are Critical Numbers?
Definition
A critical number (or critical point) of a function is a value of x within the domain of the function where the derivative is either zero or undefined. These points are significant because they indicate possible locations where the function could have:
- A local maximum
- A local minimum
- An inflection point
Importance in Calculus
Identifying critical numbers is a crucial step in the first derivative test and second derivative test, which help determine the nature of the critical points. They are also instrumental in solving optimization problems, where the goal is to find the maximum or minimum value of a function within a certain interval.
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Step-by-Step Guide to Find Critical Numbers of F(x)=25x²
Let's delve into how to find the critical numbers of the specific function:
\[ F(x) = 25x^2 \]
Step 1: Find the derivative of F(x)
The first step in locating critical numbers is to differentiate the function with respect to x.
Given:
\[ F(x) = 25x^2 \]
Using basic differentiation rules, namely the power rule:
\[ \frac{d}{dx} [ax^n] = a n x^{n-1} \]
we get:
\[ F'(x) = 25 \times 2 x^{2 - 1} = 50x \]
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Step 2: Set the derivative equal to zero and solve for x
To find critical points, solve:
\[ F'(x) = 0 \]
Substituting the derivative:
\[ 50x = 0 \]
Divide both sides by 50:
\[ x = 0 \]
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Step 3: Check for points where the derivative is undefined
Since the derivative \( F'(x) = 50x \) is a polynomial function, it is defined for all real numbers. Therefore, there are no points where the derivative is undefined.
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Critical Numbers for F(x) = 25x²
Based on the above steps, the critical number occurs where the derivative equals zero:
\[ x = 0 \]
And since the derivative is defined everywhere, no other critical points exist.
Answer:
\[
\boxed{0}
\]
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Understanding the Critical Number in Context
Graphical Interpretation
The function \( F(x) = 25x^2 \) is a parabola opening upwards, with its vertex at the origin (0, 0). The critical number at \( x=0 \) corresponds to this vertex, which is the global minimum of the function.
Significance
- At \( x=0 \), the slope of the tangent to the curve is zero.
- The function attains its minimum value at this point, which is:
\[
F(0) = 25 \times 0^2 = 0
\]
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Additional Concepts Related to Critical Numbers
Critical Numbers and Extrema
While critical numbers are potential locations for local maxima or minima, not every critical point necessarily corresponds to an extremum. To determine whether a critical point is a maximum, minimum, or neither, additional tests are used:
- First Derivative Test: Analyze the sign changes of \( F'(x) \) around the critical point.
- Second Derivative Test: Examine \( F''(x) \) at the critical point.
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Applying the Second Derivative Test to \( F(x)=25x^2 \)
Let's compute the second derivative:
\[ F''(x) = \frac{d}{dx} (F'(x)) = \frac{d}{dx} (50x) = 50 \]
Since \( F''(x) = 50 > 0 \), the function is concave up everywhere, indicating that the critical point at \( x=0 \) is a local minimum.
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Summary of Key Points
| Aspect | Details |
|---------|---------|
| Function | \( F(x) = 25x^2 \) |
| Derivative | \( F'(x) = 50x \) |
| Critical Number | \( x=0 \) |
| Derivative undefined? | No, derivative is defined everywhere |
| Nature of critical point | Global minimum at \( x=0 \) |
| Function value at critical point | \( F(0) = 0 \) |
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Practical Applications of Finding Critical Numbers
Understanding how to find critical points has numerous real-world applications, including:
- Optimization problems: Maximizing profits or minimizing costs.
- Physics: Finding points of equilibrium in motion or energy functions.
- Engineering: Structural analysis where maximum stress points are critical.
For example, in designing a roller coaster, engineers need to find the critical points of the height function to ensure safety and thrill.
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Additional Practice Problems
To reinforce your understanding, consider solving the following:
- Find the critical numbers of \( G(x) = -4x^3 + 12x \).
- Determine the critical points of \( H(x) = \frac{1}{3}x^3 - 3x + 5 \).
- For the function \( J(x) = \sqrt{x} \), identify the critical points within its domain.
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Conclusion
In this comprehensive guide, we've demonstrated how to find the critical numbers of the function \( F(x) = 25x^2 \). The process involves differentiating the function, setting the derivative equal to zero, and solving for x. For this particular quadratic function, the only critical number is at \( x=0 \), corresponding to the vertex of the parabola, which represents a global minimum. Recognizing and analyzing critical points is an essential skill in calculus, with broad applications across various fields.
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Final Note
Always remember that critical numbers are just the starting point in analyzing a function's behavior. To fully understand the nature of these points, employ the first and second derivative tests, and consider the function's context and graph.
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By mastering the process of finding critical numbers, you enhance your calculus problem-solving toolkit, enabling you to analyze and optimize functions with confidence.
