Problem 76
Question
In Exercises \(75-80,\) you will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$ f(x)=-x^{4}+4 x^{3}-4 x+1, \quad[-3 / 4,3] $$
Step-by-Step Solution
Verified Answer
Evaluate Critical Points and Endpoints to determine extrema. Use CAS for plotting and solving derivative equations.
1Step 1: Plotting the Function
Use any computer algebra system (CAS) to plot the function \( f(x) = -x^4 + 4x^3 - 4x + 1 \) over the interval \([ -\frac{3}{4}, 3] \). Observe the general behavior to help visualize where extrema might occur.
2Step 2: Finding Critical Points from f' = 0
Calculate the derivative \( f'(x) = -4x^3 + 12x^2 - 4 \). To find where \( f'(x) = 0 \), solve the equation \(-4x^3 + 12x^2 - 4 = 0\). Factor or use a numerical solver if necessary to approximate solutions.
3Step 3: Finding Critical Points from f' Not Existing
For polynomial functions, the derivative \( f' \) exists everywhere. Thus, there are no points in the interval where \( f' \) does not exist.
4Step 4: Evaluating the Function at Critical Points and Endpoints
Using the critical points found from \( f'(x) = 0 \), evaluate the original function \( f(x) \) at these points. Also, evaluate \( f(x) \) at the endpoints \( x = -\frac{3}{4} \) and \( x = 3 \).
5Step 5: Identifying Absolute Extrema
Compare all the function values obtained in Step 4. The largest value found is the absolute maximum, and the smallest value is the absolute minimum. Determine the x-values where these occur.
Key Concepts
Critical PointsDerivativePolynomial Function Analysis
Critical Points
Critical points are pivotal in the study of calculus extrema because they help us locate where a function may have local maximum or minimum values within a defined interval.
They are the points on a graph where the derivative of the function is zero or undefined. These are possible locations for the function to change its direction or shape.
To identify critical points, you need to find where the function's derivative equates to zero. For a polynomial function like the one given, you must calculate the derivative and solve the resulting equation.
In our example, the derivative is \( f'(x) = -4x^3 + 12x^2 - 4 \). By setting this equal to zero, you can solve for the variable \( x \) to find potential critical points where the slope of the tangent to the curve is horizontal.
Since our function is a polynomial, the derivative exists everywhere; therefore, the focus was only on finding points where \( f'(x) = 0 \). No critical points arise from the derivative not existing in this scenario.
They are the points on a graph where the derivative of the function is zero or undefined. These are possible locations for the function to change its direction or shape.
To identify critical points, you need to find where the function's derivative equates to zero. For a polynomial function like the one given, you must calculate the derivative and solve the resulting equation.
In our example, the derivative is \( f'(x) = -4x^3 + 12x^2 - 4 \). By setting this equal to zero, you can solve for the variable \( x \) to find potential critical points where the slope of the tangent to the curve is horizontal.
Since our function is a polynomial, the derivative exists everywhere; therefore, the focus was only on finding points where \( f'(x) = 0 \). No critical points arise from the derivative not existing in this scenario.
Derivative
The derivative of a function is a fundamental concept in calculus that determines the rate of change of that function.
In simpler terms, it tells you how steep the slope of the function’s graph is at any given point. This is why finding derivatives is crucial when analyzing polynomial functions.
In the exercise, we found the derivative \( f'(x) = -4x^3 + 12x^2 - 4 \) to understand how the original function behaves regarding its incline and decline within the specified interval.
Derivatives help us determine critical points, but they also give insights on the function's increasing or decreasing nature. For example, if \( f'(x) > 0 \), the function is increasing; if \( f'(x) < 0 \), it is decreasing.
When solving for where the derivative equals zero, it is often helpful to factor the derivative or use a computer algebra system (CAS) to approximate solutions. These solutions then help locate the critical points needed to evaluate the function further for absolute extrema.
In simpler terms, it tells you how steep the slope of the function’s graph is at any given point. This is why finding derivatives is crucial when analyzing polynomial functions.
In the exercise, we found the derivative \( f'(x) = -4x^3 + 12x^2 - 4 \) to understand how the original function behaves regarding its incline and decline within the specified interval.
Derivatives help us determine critical points, but they also give insights on the function's increasing or decreasing nature. For example, if \( f'(x) > 0 \), the function is increasing; if \( f'(x) < 0 \), it is decreasing.
When solving for where the derivative equals zero, it is often helpful to factor the derivative or use a computer algebra system (CAS) to approximate solutions. These solutions then help locate the critical points needed to evaluate the function further for absolute extrema.
Polynomial Function Analysis
Analyzing a polynomial function involves a few different steps to completely understand its behavior over a certain interval.
Key tasks include finding critical points, evaluating the function at these points, and understanding its overall trend.A polynomial function is an expression that consists of variables and coefficients, all with whole number exponents, like \( f(x) = -x^4 + 4x^3 - 4x + 1 \).
To analyze such functions, you typically compute the first derivative, as in the exercise, which can be used to determine the critical points for potential local maxima and minima.
The process involves:
Key tasks include finding critical points, evaluating the function at these points, and understanding its overall trend.A polynomial function is an expression that consists of variables and coefficients, all with whole number exponents, like \( f(x) = -x^4 + 4x^3 - 4x + 1 \).
To analyze such functions, you typically compute the first derivative, as in the exercise, which can be used to determine the critical points for potential local maxima and minima.
The process involves:
- Finding the derivative and simplifying it, if possible.
- Setting the derivative equal to zero to find critical points.
- Evaluating the original function at these critical points and at the interval endpoints.
Other exercises in this chapter
Problem 75
Suppose the derivative of the function \(y=f(x)\) is $$ y^{\prime}=(x-1)^{2}(x-2) $$ At what points, if any, does the graph of \(f\) have a local minimum, local
View solution Problem 76
Solve the initial value problems in Exercises \(67-86\). $$ \frac{d r}{d \theta}=\cos \pi \theta, \quad r(0)=1 $$
View solution Problem 76
Suppose the derivative of the function \(y=f(x)\) is $$ y^{\prime}=(x-1)^{2}(x-2)(x-4) $$ At what points, if any, does the graph of \(f\) have a local minimum,
View solution Problem 77
Solve the initial value problems in Exercises \(67-86\). $$ \frac{d v}{d t}=\frac{1}{2} \sec t \tan t, \quad v(0)=1 $$
View solution