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Magazine Chapter 24: Problem 32
In Exercises \(21-32,\) sketch the graphs of the given functions by determining the appropriate information and points from the first and second derivatives. Use a calculator to check the graph. In Exercises \(27-32,\) use the function maximum-minimum feature to check the local maximum and minimum points. $$y=x^{4}+32 x+2$$
Short Answer
Expert verified
The local minimum occurs at \( x = -2 \) with the function concave up at this point.
Step by step solution
01
Find the First Derivative
The first derivative of the function gives us information about the slope of the tangent line at any point and can indicate where the function is increasing or decreasing. For the function \( y = x^4 + 32x + 2 \), the first derivative is found using the power rule: \( \frac{dy}{dx} = 4x^3 + 32 \).
02
Critical Points from the First Derivative
Critical points occur where the first derivative is zero or undefined. To find where \( 4x^3 + 32 = 0 \), solve for \( x \): \( 4x^3 = -32 \) which simplifies to \( x^3 = -8 \), giving \( x = -2 \). This is our critical point.
03
Find the Second Derivative
The second derivative gives us information about the concavity of the function and can help identify inflection points. The second derivative of \( y = x^4 + 32x + 2 \) is \( \frac{d^2y}{dx^2} = 12x^2 \).
04
Analyze the Second Derivative
Evaluate the second derivative at the critical point, \( x = -2 \). We have \( 12(-2)^2 = 48 \), which is positive, indicating that the function is concave up and \( x = -2 \) is a local minimum point.
05
Sketch the Graph and Verify with a Calculator
Use the critical point information and calculate a few more points if needed for accuracy. Sketch the curve showing that the function is decreasing to the left of \( x = -2 \) and increasing to the right with a local minimum at \( x = -2 \). Verify with a graphing calculator to ensure the shape and critical points match.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
First Derivative
The first derivative of a function is a powerful tool in calculus for understanding how a function behaves. In simple terms, it gives us the slope of the tangent line at any given point along the function's curve. This slope can tell us if the function is increasing or decreasing at that specific point.
For our function, which is given as \( y = x^4 + 32x + 2 \), we used the power rule to find the first derivative: \( \frac{dy}{dx} = 4x^3 + 32 \).
For our function, which is given as \( y = x^4 + 32x + 2 \), we used the power rule to find the first derivative: \( \frac{dy}{dx} = 4x^3 + 32 \).
- Where the derivative is positive, the function is increasing.
- Where the derivative is negative, the function is decreasing.
- A zero derivative indicates a potential maximum, minimum, or saddle point.
Second Derivative
The second derivative of a function gives us deeper insight, particularly into its concavity鈥攁nd from this, essential characteristics such as inflection points and local extrema. Derived from the first derivative by further differentiation, the second derivative helps pinpoint how the function's rate of change is itself changing.
For the function \( y = x^4 + 32x + 2 \), the second derivative is \( \frac{d^2y}{dx^2} = 12x^2 \).
For the function \( y = x^4 + 32x + 2 \), the second derivative is \( \frac{d^2y}{dx^2} = 12x^2 \).
- When the second derivative is positive, the function is concave upwards like a smile, meaning it increases at an increasing rate.
- When the second derivative is negative, the function is concave downwards like a frown.
- If the second derivative equals zero, the function could have an inflection point, which is where the concavity changes.
Critical Points
Critical points are special x-values where the function's first derivative equals zero or does not exist. These points are vital because they tell us about changes in the function's behavior and help in identifying potential maxima, minima, or inflection points.
For our example function, \( 4x^3 + 32 = 0 \) simplifies to \( x^3 = -8 \), resulting in \( x = -2 \) as a critical point. Here, the derivative is zero, suggesting a possible turnaround point or extremum.
Critical points can:
For our example function, \( 4x^3 + 32 = 0 \) simplifies to \( x^3 = -8 \), resulting in \( x = -2 \) as a critical point. Here, the derivative is zero, suggesting a possible turnaround point or extremum.
Critical points can:
- Indicate high and low spots in the curve, known as local maxima and minima.
- Signal changes in direction, such as moving from increasing to decreasing, and vice versa.
Local Minimum
A local minimum refers to a point where a function has its lowest value in the surrounding interval. Identifying a local minimum involves combining the information from both the first and second derivatives.
From our analysis of the function \( y = x^4 + 32x + 2 \), we determined that the critical point \( x = -2 \) is a local minimum.
From our analysis of the function \( y = x^4 + 32x + 2 \), we determined that the critical point \( x = -2 \) is a local minimum.
- The first derivative equals zero at this point, pointing to a potential extremum.
- The second derivative is positive, indicating the function is concave up here, akin to a bowl shape. Thus, \( x = -2 \) is a local minimum.
Function Concavity
Function concavity tells us more than just the direction the function curves. It adds a layer of depth, predicting the 'shape' of the graph and how the function accelerates or decelerates in escalation or descent.
Using the second derivative \( \frac{d^2y}{dx^2} = 12x^2 \), we assessed that it is always positive for all real x-values, indicating the function is concave up everywhere.
Using the second derivative \( \frac{d^2y}{dx^2} = 12x^2 \), we assessed that it is always positive for all real x-values, indicating the function is concave up everywhere.
- When concave up, a function resembles a U-shape, implying that as the x-values increase, the function's rate of increase is itself increasing.
- Understanding concavity assists in better visualizing the graph and anticipating the function's behavior in broader terms.
