Question

A function \(f(x)\) is given. Find the critical points of \(f\) and use the Second Derivative Test, when possible, to determine the relative extrema. \(f(x)=-x^{2}-5 x+7\)

Short answer

The function has a local maximum at \(x = -\frac{5}{2}\).

Step-by-step solution

Find the Derivative

First, we need to find the first derivative of the function \(f(x) = -x^2 - 5x + 7\). The derivative is found using the power rule, where the derivative of \(ax^n\) is \(n \cdot ax^{n-1}\). Thus, \[ f'(x) = \frac{d}{dx}(-x^2) + \frac{d}{dx}(-5x) + \frac{d}{dx}(7) = -2x - 5. \]

Find Critical Points

Critical points occur where the first derivative is equal to zero. Set \(f'(x) = 0\) and solve for \(x\): \[ -2x - 5 = 0 \] Solve for \(x\): \[ -2x = 5 \] \[ x = -\frac{5}{2}. \] Therefore, the critical point is \(x = -\frac{5}{2}\).

Use the Second Derivative Test

Next, find the second derivative to apply the Second Derivative Test, which helps determine the nature of the critical point. The second derivative \(f''(x)\) is the derivative of \(f'(x)\): \[ f''(x) = \frac{d}{dx}(-2x - 5) = -2. \] The Second Derivative Test states that if \(f''(x) > 0\), the point is a local minimum; if \(f''(x) < 0\), the point is a local maximum. Since \(f''(x) = -2 < 0\), the critical point \(x = -\frac{5}{2}\) is a local maximum.

Key concepts

Second Derivative Test

The Second Derivative Test is a mathematical tool used to determine the nature of critical points in a function. Critical points are where the function’s rate of change (the first derivative) is zero, and the second derivative helps us understand whether these points are local maxima, local minima, or neither.

The key steps to applying the Second Derivative Test include:

• Finding the first derivative of the function.
• Solving the equation where the first derivative equals zero to find critical points.
• Calculating the second derivative of the function.

Once we have the second derivative, we analyze its value at each critical point:

• If the second derivative is positive ( \( f''(x) > 0 \) ), the function is concave up at that point, indicating a local minimum.
• If the second derivative is negative ( \( f''(x) < 0 \) ), the function is concave down, indicating a local maximum.
• If the second derivative equals zero, the test is inconclusive, and other methods may be needed to determine the point's nature.

Using this test simplifies the process of identifying whether a function's critical points are peaks, valleys, or neither.

First Derivative

The first derivative of a function, often denoted as \( f'(x) \), represents the rate of change or slope at any given point on the function. In essence, it describes how the function's output changes with respect to changes in the input.

For polynomial functions, finding the first derivative employs the power rule, which states that the derivative of \( a x^n \) is \( n \, a x^{n-1} \). This rule makes it straightforward to differentiate polynomials.

The importance of the first derivative lies in its ability to locate critical points, where the function's slope is zero. These are the points where potential maxima or minima might occur. It's a crucial step in analyzing the function's behavior because:

• Setting \( f'(x) = 0 \) gives necessary conditions for any relative extrema.
• It helps identify whether the function is increasing or decreasing at certain intervals.

Understanding the first derivative is fundamental to identifying and classifying the critical points of a function.

Relative Extrema

Relative extrema refer to the local maximum and minimum values of a function. These are points where the function either peaks or dips, compared to its immediate surroundings.

In a mathematical sense, a function \( f(x) \) has a local maximum at \( x = c \) if \( f(c) \) is greater than \( f(x) \) for values of \( x \) near \( c \). Conversely, \( f(x) \) has a local minimum at \( x = c \) if \( f(c) \) is less than \( f(x) \) for nearby \( x \).

Critical points identified by setting the first derivative to zero are potential candidates for relative extrema. To conclusively determine the nature of these points, we apply tests like the Second Derivative Test. Here’s what we do with those points:

• Use the Second Derivative Test to ascertain whether the point is a local maximum or minimum.
• Determine whether further investigation is needed if the second derivative is zero.

Understanding relative extrema is vital as they indicate important trends in the graph of the function, implying significant increases or decreases.

Polynomial Functions

Polynomial functions are mathematical expressions involving terms consisting of variables raised to whole number powers and their coefficients. They form a foundational concept in algebra and calculus and appear frequently in various applications.

A typical polynomial function is expressed as:
\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \]where \( a_n, a_{n-1}, ..., a_0 \) are constants known as coefficients, and \( n \) is a non-negative integer representing the degree of the polynomial.

Important properties of polynomial functions include:

• Smooth and continuous: Polynomials have no gaps or sharp corners.
• Degree determines behavior: The highest degree term dictates the polynomial's end behavior and the number of times the graph changes direction.

When analyzing polynomial functions, derivatives play an essential role in understanding their characteristics. For example, they help identify critical points, relative extrema, and intervals of increase and decrease.