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Chapter 12: Problem 21

Find the exact location of all the relative and absolute extrema of each function. \(f(x)=x^{4}-4 x^{3}\) with domain \([-1,+\infty)\)

Short Answer

Expert verified
The function \(f(x)=x^{4}-4 x^{3}\) has a relative maximum at \((0, 0)\), an absolute maximum at \((-1, 5)\), and an absolute minimum at \((3, -27)\) within the domain \([-1,+\infty)\).

Step by step solution

01

Find the first derivative

To begin, we'll find the first derivative of the function, \(f(x)=x^{4}-4 x^{3}\). Using the power rule, we find that: \[f'(x) = \frac{d}{dx}(x^{4}-4 x^{3}) = 4x^3 - 12x^2\]
02

Find the critical points

Now, we'll find the critical points by solving the equation \(f'(x) = 0\). We have: \[4x^3 - 12x^2 = 0\] Factoring out the common factor of 4x^2, we get: \[4x^2(x - 3) = 0\] This gives us the critical points, \(x = 0\) and \(x = 3\).
03

Evaluate endpoints

The given domain is \([-1, +\infty)\). The only endpoint is \(x=-1\). We evaluate \(f(x)\) at this endpoint: \[f(-1)=(-1)^{4}-4(-1)^{3}=1+4=5\]
04

Determine relative extrema

To determine the relative extrema, we'll analyze the intervals formed by the critical points. We have three intervals to test, due to the two critical points: \((-\infty, 0)\), \((0, 3)\), and \((3, \infty)\). Testing each interval with a point within its bounds, we have: - Inteval \((-\infty, 0)\): choose \(x=-2\); \(f'(-2)=64-48>0\), meaning that \(f(x)\) is increasing on this interval. - Inteval \((0, 3)\): choose \(x=1\); \(f'(1)=4-12<0\), meaning that \(f(x)\) is decreasing on this interval. - Inteval \((3, \infty)\): choose \(x=4\); \(f'(4)=256-192>0\), meaning that \(f(x)\) is increasing on this interval. This means that \(x=0\) is a relative maximum and \(x=3\) is a relative minimum.
05

Identify absolute extrema

There are no other endpoints to check, so we'll just compare the values of the function at the critical points and the endpoint. The function takes the following values at those points: - At \(x=-1\): \(f(-1)=5\) - At \(x=0\): \(f(0)=0\) - At \(x=3\): \(f(3)=3^4-4\cdot3^3=-27\) This tells us that the absolute maximum is \( 5\) at \(x=-1\), and the absolute minimum is \(-27\) at \(x=3\). So the exact location of all relative and absolute extrema is a relative maximum at \((0, 0)\), an absolute maximum at \((-1, 5)\), and an absolute minimum at \((3, -27)\).

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Key Concepts

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

Critical Points
When studying the extrema, or high and low points, of a function, the concept of critical points is essential. These points are where the first derivative of the function equals zero or does not exist.

To find the critical points, we calculate the first derivative, set it equal to zero, and solve for the independent variable. In the problem given, the first derivative of the function, f'(x) = 4x^3 - 12x^2, is set to zero, yielding two critical points: x = 0 and x = 3.

Why do we care about critical points? They help us determine where a function could change from increasing to decreasing or vice versa, which are typically where the relative extrema are located. But remember, not all critical points are extrema. The usefulness of critical points becomes evident when we apply the first derivative test.
First Derivative Test
The first derivative test is a way to determine if a critical point is a relative minimum, relative maximum, or neither. After finding the critical points, we test the sign of the first derivative on either side of each point. The resulting change in sign can tell us the nature of the extrema.

In our function f(x) = x^4 - 4x^3, we found that f'(x) changes from positive to negative at x = 0, indicating a relative maximum, and from negative to positive at x = 3, indicating a relative minimum. The end behavior also needs to be considered by checking the first derivative beyond the critical points, which can tell us if we have absolute extrema at the boundaries of the domain.
Power Rule Differentiation
The power rule for differentiation is a basic tool used to find the derivative of functions in the form of x^n, where n is any real number. The rule states that the derivative of x^n is nx^{n-1}.

To apply the power rule to our function, f(x) = x^4 - 4x^3, we differentiate each term separately, bringing down the exponent as a coefficient, and reducing the exponent by one. The resulting first derivative, f'(x) = 4x^3 - 12x^2, demonstrates the power rule in action.

This process is vital for finding the first derivative efficiently, which is then used to calculate critical points and apply the first derivative test for identifying extrema of the function.

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Most popular questions from this chapter

Daily oil production in Mexico and daily U.S. oil imports from Mexico during \(2005-2009\) can be approximated by $$\begin{array}{ll}P(t)=3.9-0.10 t \text { million barrels } & (5 \leq t \leq 9) \\ I(t)=2.1-0.11 t \text { million barrels } & (5 \leq t \leq 9) \end{array}$$ where \(t\) is time in years since the start of \(2000 .^{41}\) Graph the function \(I(t) / P(t)\) and its derivative. Is the graph of \(I(t) / P(t)\) concave up or concave down? The concavity of \(I(t) / P(t)\) tells you that: (A) The percentage of oil produced in Mexico that was exported to the United States was decreasing. (B) The percentage of oil produced in Mexico that was not exported to the United States was increasing. (C) The percentage of oil produced in Mexico that was exported to the United States was decreasing at a slower rate. (D) The percentage of oil produced in Mexico that was exported to the United States was decreasing at a faster rate.

The approximate value of subprime (normally classified as risky) mortgage debt outstanding in the United States can be approximated by $$A(t)=\frac{1,350}{1+4.2(1.7)^{-t}} \text { billion dollars } \quad(0 \leq t \leq 8)$$ \(t\) years after the start of \(2000 .{ }^{38}\) Graph the function as well as its first and second derivatives. Determine, to the nearest whole number, the values of \(t\) for which the graph of \(A\) is concave up and concave down, and the \(t\) -coordinate of any points of inflection. What does the point of inflection tell you about subprime mortgages?

The consumer demand equation for tissues is given by \(q=(100-p)^{2}\), where \(p\) is the price per case of tissues and \(q\) is the demand in weekly sales. a. Determine the price elasticity of demand \(E\) when the price is set at \(\$ 30\), and interpret your answer. b. At what price should tissues be sold in order to maximize the revenue? c. Approximately how many cases of tissues would be demanded at that price?

. If the graph of a function has a vertical asymptote at \(x=a\) in such a way that \(y\) increases to \(+\infty\) as \(x \rightarrow a\), what can you say about the graph of its derivative? Explain.

A function is bounded if its entire graph lies between two horizontal lines. Can a bounded function have vertical asymptotes? Can a bounded function have horizontal asymptotes? Explain.
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