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Chapter 11: Problem 11

Without using a graphing calculator, sketch the following graphs. Label all local maxima and minima. Beside the sketch of \(f\), draw a rough sketch of \(f^{\prime}(x)\). (a) \(f(x)=x(x-3)(x+5)\) (b) \(f(x)=-2 x(x-3)(x+5)\) (c) \(f(x)=x^{3}+3 x^{2}-9 x\) (d) \(f(x)=x^{3}+3 x^{2}-9 x+1\)

Short Answer

Expert verified
The solutions are based on identifying roots and minima/maxima points of the functions to determine the shape of the function and its derivative.

Step by step solution

01

(a) Finding Roots and Derivative of \(f(x)=x(x-3)(x+5)\)

Find the roots by setting \(f(x) = 0\): \(x(x-3)(x+5)=0\). The roots of this equation are \(x = 0, x = 3, x = -5\). Next, differentiate the function \(f(x)\). Its derivative \(f'(x) = 3x^2 - 4x - 15\).
02

(b) Drawing \(f(x)\) and \(f'(x)\)

First draw the graph of \(f(x)\) highlighting the roots \(x = 0, x = 3, x = -5\), which are the x-intercepts of the graph. The curves slope upwards on each of the ends and delve downwards across x=0, making it a cubic function. For \(f'(x)\), plot the equation \(f'(x) = 3x^2 - 4x - 15\). Pay close attention to points where \(f'(x) = 0\) which represent local minima and maxima of the \(f(x)\) graph.
03

(c) Finding Roots and Derivative of \(f(x)=-2x(x-3)(x+5)\)

Here \(f(x) = 0\) at \(x = 0, x = 3, x = -5\). Differentiate to get \(f'(x)=-6x^2 + 20x\).
04

(d) Drawing \(f(x)\) and \(f'(x)\)

Base the sketch of \(f(x)\) off part (a), but flip it upside down since it's multiplied by -2. Then plot \(f'(x)=-6x^2 + 20x\), checking for the points where \(f'(x) = 0\).
05

(e) Finding Roots and Derivative of \(f(x)=x^3 + 3x^2 - 9x\)

Calculate the roots by solving \(f(x) = 0\), giving you \(x = 0, x = -3, x = 3\). The derivative is then \(f'(x) = 3x^2 + 6x - 9\).
06

(f) Drawing \(f(x)\) and \(f'(x)\)

First plot \(f(x)=x^3 + 3x^2 - 9x\) with the roots \(x = 0, x = -3, x = 3\). Plot \(f'(x) = 3x^2 + 6x - 9\), identifying local minima and maxima by places where \(f'(x) = 0\).
07

(g) Finding Roots and Derivative of \(f(x)=x^3 + 3x^2 - 9x + 1\)

The roots of this equation can't be simply found by factoring. The derivative is still \(f'(x) = 3x^2 + 6x - 9\).
08

(h) Drawing \(f(x)\) and \(f'(x)\)

Sketch the curve \(f(x)=x^3 + 3x^2 - 9x + 1\) similarly to the graph in part (e), but adjust the y-intercept by one unit up to 1. For \(f'(x)\), plot \(3x^2 + 6x - 9\) similarly to the graph in part (f).

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Key Concepts

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

Finding Roots
Roots of a polynomial are special values where the polynomial equals zero. They are also known as the x-intercepts because they indicate where the graph crosses the x-axis. Finding these roots involves solving the equation set by making the polynomial function zero, i.e., setting \( f(x) = 0 \).
  • For example, in the function \( f(x) = x(x-3)(x+5) \), you find roots by solving \( x(x-3)(x+5) = 0 \). This gives the roots \( x = 0, x = 3, \) and \( x = -5 \).
  • Each factor in the polynomial, when set to zero, offers a root of the equation.
  • Roots help us understand key features of the graph, including where it might snack above and below the x-axis.
Understanding these roots is crucial because they provide a basic framework for sketching the graph of a polynomial function.
Derivatives
Derivatives in calculus give us the rate at which a function is changing at any given point. This is particularly useful in graphing, as the derivative tells us about the slope of the function at different points.
  • The first derivative, \( f'(x) \), gives insight into the function's increasing or decreasing behavior.
  • For \( f(x) = x(x-3)(x+5) \), the derivative is \( f'(x) = 3x^2 - 4x - 15 \). This derivative is calculated using the rules of differentiation.
  • By solving \( f'(x) = 0 \), we find critical points which are essential in identifying local maximums and minimums.
Understanding derivatives will aid in sketching the curves accurately and anticipating the function's behavior at key intervals of the x-axis.
Local Maxima and Minima
Local maxima and minima are points where the function reaches a peak (maximum) or a trough (minimum) within a certain region. These points are important for sketching because they indicate changes in the direction of the graph.
  • To find these points, you utilize the derivative \( f'(x) \).
  • Set \( f'(x) = 0 \) and solve for \( x \) to get the critical points.
  • Using a sign test or the second derivative test, determine whether each critical point is a maxima or minima.
Identifying these key turning points on the graph helps define the overall shape and character of the polynomial's graph.
Polynomial Functions
Polynomial functions are expressions composed of variables and coefficients, utilizing operations like addition, subtraction, multiplication, but where no division by a variable occurs. They are usually expressed in the form \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \,\ldots\, + a_1 x + a_0 \).
  • These functions can exhibit various behaviors due to their degree, the highest power of \( x \).
  • A third-degree polynomial like \( f(x) = x(x-3)(x+5) \) will generally have up to three real roots and at most two turning points.
  • Understanding the structure of polynomial functions helps in sketching their graphs accurately, predicting their end-behaviors, and knowing how they can be factored to find roots.
Polynomials are a fundamental part of algebra and calculus and have wide applications in mathematical modeling and problem-solving.

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Most popular questions from this chapter

Find the (real) zeros of the polynomial given. (a) \(f(x)=-x^{3}-x^{2}-5 x\) (b) \(g(x)=0.5 x^{4}-0.5\)

Graph \(f(x)=\frac{x^{4}}{4}+x^{3}-5 x^{2}\). Indicate the \(x\) -coordinates of all local extrema and all points of inflection. What is the absolute minimum value of \(f ?\) The absolute maximum value?

Give an example of a cubic function \(f(x)\) with the characteristic(s) specified. Your answer should be a formula, but a picture will be helpful. There may be many possible answers. \(f(x)\) has zeros at \(x=-1\) and \(x=2\) only. \(f(0)=1\).

\(P(x)\) is a polynomial with the characteristics specified. For each statement following the characteristic, determine whether the statement is definitely true; possibly true, but not necessarily true; or definitely false. Explain. The graph of \(P(x)\) is symmetric about the origin. (a) The degree of \(P(x)\) is even. (b) The degree of \(P(x)\) is odd. (c) \(P(x)\) has no zeros. (d) \(P(x)\) has at least one zero. (e) The number of turning points of \(P(x)\) is of the form \(2 n\) where \(n\) is a nonnegative integer.

Let \(f(x)=(x-a)(x-b)^{2}\), where \(a>b>0\). By looking at the sign of \(f\) you can show that \(f\) has a local maximum at \(x=b\). This problem asks you to verify this using the second derivative test. (a) Using the Product Rule, show \(f^{\prime}(b)=0\). (b) Use the second derivative test to show that \(f\) has a local maximum at \(x=b\).
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