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Chapter 4: Problem 49

Find the absolute maximum and absolute minimum values of \(f\) on the given interval. \(f(x)=2 x^{3}-3 x^{2}-12 x+1,[-2,3]\)

Short Answer

Expert verified
Max: 8 at x=-1; Min: -19 at x=2.

Step by step solution

01

Find Critical Points

First, we need to find the critical points of the function where the derivative is zero or undefined. Find the derivative of the function:\[f'(x) = 6x^2 - 6x - 12\]Set the derivative equal to zero to find critical points:\[6x^2 - 6x - 12 = 0\]Divide the entire equation by 6 to simplify:\[x^2 - x - 2 = 0\]Factor the quadratic equation:\[(x - 2)(x + 1) = 0\]The solutions to this equation are the critical points: \[x = 2 \text{ and } x = -1\]
02

Evaluate Function at Critical Points and Endpoints

Evaluate the function \(f(x)\) at the critical points and endpoints of the interval \([-2, 3]\).Calculate \(f(x)\) at the endpoints and critical points:1. \(f(-2) = 2(-2)^3 - 3(-2)^2 - 12(-2) + 1 = -16 - 12 + 24 + 1 = -3\)2. \(f(3) = 2(3)^3 - 3(3)^2 - 12(3) + 1 = 54 - 27 - 36 + 1 = -8\)3. \(f(-1) = 2(-1)^3 - 3(-1)^2 - 12(-1) + 1 = -2 - 3 + 12 + 1 = 8\)4. \(f(2) = 2(2)^3 - 3(2)^2 - 12(2) + 1 = 16 - 12 - 24 + 1 = -19\)
03

Determine Absolute Maximum and Minimum

From the values calculated:- \(f(-2) = -3\)- \(f(3) = -8\)- \(f(-1) = 8\)- \(f(2) = -19\)The absolute maximum value of \(f(x)\) on the interval \([-2, 3]\) is \(8\), which occurs at \(x = -1\).The absolute minimum value of \(f(x)\) on the interval \([-2, 3]\) is \(-19\), which occurs at \(x = 2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Points
Critical points are where a function's derivative is either zero or undefined. They are crucial in finding the points of interest on a graph, such as local maxima or minima. To find these points mathematically:
  • Take the derivative of the function.
  • Set the derivative equal to zero: solve for the variable.
  • Find where the derivative is undefined, if applicable.
For the given function, the derivative is \(f'(x)=6x^2 - 6x - 12\). Setting it to zero gives \(6x^2 - 6x - 12 = 0\). From here, we simplify to find critical points \(x = 2\) and \(x = -1\). These values help in determining absolute maxima, minima, or neither, depending on the context.
Absolute Maximum
An absolute maximum is the highest value that a function reaches on a given interval. This value, when compared across the domain, stands above all. To find the absolute maximum:
  • Calculate the function's value at each critical point.
  • Evaluate the function at the endpoints of the interval.
  • Compare all these values to identify the maximum.
In the problem, we evaluated the function at \(x = -2, 3, -1,\) and \(2\). Among these, \(f(-1) = 8\) is the largest value. Thus, the absolute maximum is \(8\), occurring at \(x = -1\).
Absolute Minimum
The absolute minimum is the opposite of the absolute maximum, indicating the lowest value that a function achieves in its domain. Finding the absolute minimum involves a similar process as finding the maximum:
  • Evaluate the function at each critical point and at the interval's endpoints.
  • Compare these values to find the smallest one.
For the provided function, \(f(x)\) gives values \(-3, -8, 8,\) and \(-19\). The smallest among these is \(-19\), occurring at \(x = 2\), making it the absolute minimum.
Derivative
The derivative of a function measures how the function's output value changes as the input changes. It's often represented as \(f'(x)\) or \(\frac{dy}{dx}\). Derivatives are used to find the slope of the tangent line at any point on a curve:
  • Positive derivative: function is increasing.
  • Negative derivative: function is decreasing.
  • Zero derivative: potential maximum/minimum, known as critical points.
For the function \(f(x)=2x^3-3x^2-12x+1\), the derivative is calculated as \(f'(x)=6x^2-6x-12\). This derivative helps in identifying where the function reaches potential maxima or minima.
Factor Quadratic
Factoring quadratics is a key algebraic technique for solving quadratic equations, especially useful in finding critical points in calculus. A quadratic in the form \(ax^2+bx+c\) can often be factored into \((mx+n)(px+q)\). Here's how to factor:
  • Write the equation in standard form.
  • Find two numbers that multiply to the constant term \(a\cdot c\) and add to the linear coefficient \(b\).
  • Use these numbers to split the middle term and factor by grouping.
  • Solve the factored equation to find the values of \(x\).
For the derivative \(x^2-x-2=0\), it factors into \((x-2)(x+1)=0\), yielding solutions \(x=2\) and \(x=-1\), essential in identifying critical points and further analyses.

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Most popular questions from this chapter

The manager of a 100 -unit apartment complex knows from experience that all units will be occupied if the rent is \(\$ 800\) per month. A market survey suggests that, on average, one additional unit will remain vacant for each \(\$ 10\) increase in rent. What rent should the manager charge to maximize revenue?

Describe how the graph of \(f\) varies as \(c\) varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also identify any transitional values of \(c\) at which the basic shape of the curve changes. \(f(x)=\frac{c x}{1+c^{2} x^{2}}\)

(a) Apply Newton's method to the equation \(1 / x-a=0\) to derive the following reciprocal algorithm: \(\quad x_{n+1}=2 x_{n}-a x_{n}^{2}\) This algorithm enables a computer to find reciprocals without actually dividing.) b) Use part (a) to compute 1\(/ 1.6984\) correct to six decimal places.

If \(f^{\prime}\) is continuous, use I'Hospital's Rule to show that $$\lim _{h \rightarrow 0} \frac{f(x+h)-f(x-h)}{2 h}=f^{\prime}(x)$$ Explain the meaning of this equation with the aid of a diagram.

A stone is dropped from the upper observation deck (the space Deck of the CN Tower, 450 \(\mathrm{m}\) above the ground. (a) Find the distance of the stone above ground level at time t. (b) How long does it take the stone to reach the ground? (c) With what velocity does it strike the ground? (d) If the stone is thrown downward with a speed of 5 \(\mathrm{m} / \mathrm{s}\) , how long does it take to reach the ground?
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