Problem 89 - Thomas Calculus

Question

Graph \(f(x)=2 x^{4}-4 x^{2}+1\) and its first two derivatives together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\) and \(f^{\prime \prime} .\)

Step-by-Step Solution

Answer

The function has a local max at \(x=0\) and local mins at \(x=\pm 1\), with increasing/decreasing behavior mirrored by \(f'\) and concavity by \(f''\).

Step 1: Find the First Derivative

To find the first derivative of the function, we need to differentiate it with respect to \(x\). The function given is \(f(x) = 2x^4 - 4x^2 + 1\). Differentiating term by term, we get:\[f'(x) = \frac{d}{dx}(2x^4) - \frac{d}{dx}(4x^2) + \frac{d}{dx}(1)\]\[f'(x) = 8x^3 - 8x\].

Step 2: Find the Second Derivative

Next, we differentiate the first derivative \(f'(x) = 8x^3 - 8x\) to find the second derivative of the function:\[f''(x) = \frac{d}{dx}(8x^3) - \frac{d}{dx}(8x)\]\[f''(x) = 24x^2 - 8\].

Step 3: Analyze the Critical Points

To find the critical points of \(f(x)\), set the first derivative equal to zero:\[8x^3 - 8x = 0\]Factorizing gives:\[8x(x^2 - 1) = 0\]Thus, \(x = 0\), \(x = 1\), and \(x = -1\). These are critical points. We test these in the second derivative to determine concavity.

Step 4: Determine Concavity and Inflection Points

Substitute the critical points into the second derivative to understand concavity:- For \(x = 0\), \(f''(0) = 24(0)^2 - 8 = -8\) (concave down).- For \(x = 1\), \(f''(1) = 24(1)^2 - 8 = 16\) (concave up).- For \(x = -1\), \(f''(-1) = 24(-1)^2 - 8 = 16\) (concave up).Therefore, \((0, f(0))\) is a point of local maximum and \((1, f(1))\) and \((-1, f(-1))\) are points of local minimum.

Step 5: Graph the Functions

Plot the original function, first derivative, and second derivative:- For \(f(x) = 2x^4 - 4x^2 + 1\), it is a quartic function which usually results in a "W" shape.- \(f'(x) = 8x^3 - 8x\) is a cubic function with zeros at \(-1, 0, \) and \(1\).- \(f''(x) = 24x^2 - 8\) is a parabola opening upwards with a vertex at \(x=0\).

Step 6: Interpretation of the Graphs

Compare the graphs to analyze behavior:- When \(f'(x) > 0\), \(f(x)\) is increasing; when \(f'(x) < 0\), \(f(x)\) is decreasing.- At critical points where \(f'(x) = 0\), check \(f''(x)\): if \(f''(x) > 0\) it's a local minimum, if \(f''(x) < 0\) it's a local maximum.

Key Concepts

Derivative

In calculus, the derivative is a fundamental tool used to determine the rate at which a function is changing at any point. For the function given, \(f(x) = 2x^4 - 4x^2 + 1\), we find the first derivative, which is \(f'(x) = 8x^3 - 8x\). This represents the slope of the tangent to the graph at any point \(x\).

• The rate of change tells us how quickly the function rises or falls.
• Positive derivative values indicate that \(f(x)\) is increasing, while negative values show that it is decreasing.

Understanding derivatives is crucial for finding critical points and analyzing the overall behavior of functions. The process involves differentiating each term with respect to \(x\), making it a powerful technique for examining changes.

Critical Points

Critical points in a function occur where the derivative is zero or undefined. These points often indicate where the function might have a relative maximum, minimum, or a point of inflection. For the function \(f(x) = 2x^4 - 4x^2 + 1\), the critical points are found by solving \(f'(x) = 0\), which leads to the values \(x = 0\), \(x = 1\), and \(x = -1\).

• At these points, the slope of the tangent is zero, indicating potential peaks or troughs.
• Critical points are essential for sketching accurate graphs.

By evaluating these critical points with the second derivative, further insights can be gained into the nature of these points, classifying them as maxima, minima, or points of inflection.

Concavity

Concavity describes the direction in which a curve bends, determined by the sign of the second derivative of a function. In our function, the second derivative is \(f''(x) = 24x^2 - 8\). This helps us understand the curvature of the function's graph.

• If \(f''(x) > 0\), the function is concave up, resembling a bowl shape.
• If \(f''(x) < 0\), the function is concave down, like an upside-down bowl.

Inflection points occur where the concavity changes. For our function, we found that at \(x = 0\), the function is concave down, while at \(x = 1\) and \(x = -1\), it is concave up. Identifying concavity is helpful for determining the function's behavior and graphing.

Graphing Functions

Graphing functions provides a visual representation of a mathematical relationship. By plotting \(f(x) = 2x^4 - 4x^2 + 1\) along with its first two derivatives, we observe various features such as critical points, extrema, and concavity. Graphing highlights both the dynamics of \(f(x)\) and the influence of its derivatives:

• \(f(x)\) gives us the overall shape, typically a "W" shape for quartic functions.
• \(f'(x)\) allows us to see where the function is increasing or decreasing.
• \(f''(x)\) reveals the concavity, showing upward or downward curvature.

Overall, graphing the function with derivatives aids in a comprehensive understanding of its behavior and helps us predict how the function behaves across different intervals.