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Chapter 3: Problem 33

Using the Second Derivative Test In Exercises \(33-44\) , find all relative extrema of the function. Use the Second Derivative Test where applicable.$$f(x)=6 x-x^{2}$$

Short Answer

Expert verified
The function \(f(x)=6x-x^{2}\) has a relative maxima at \(x=3\).

Step by step solution

01

Find the Derivative

Take the derivative of the function \(f(x)=6x-x^{2}\) using the power rule, which will yield \(f'(x)=6-2x\).
02

Find Critical Points

Set the derivative equal to zero and solve for \(x\) to find the critical points. You will get \(x=3\).
03

Second Derivative Test

Take the second derivative of the function, which again using the power rule, will give \(f''(x)=-2\). Because \(f''(x)<0\), this tells you that the function concaves down at \(x=3\). According to the Second Derivative Test, since the concavity is negative, \(x=3\) is a relative maxima.

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Key Concepts

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

Relative Extrema
Relative extrema refer to the points on a graph where a function reaches its local maximum or minimum values. These points are essential because they help in understanding the behavior and shape of the graph, showing where the function increases or decreases. When analyzing a function, identifying relative extrema provides insight into important features.
  • Relative Maximum: This occurs at point(s) where the function values are greater than those surrounding it, indicating a "peak."
  • Relative Minimum: Here, the function has values smaller than those around it, signifying a "valley."
  • To find these extrema, you'll often use tests like the First and Second Derivative Tests, which offer reliable methods to determine these critical points.
Knowing where these extrema exist allows for better predictions and a more comprehensive understanding of the function's dynamics.
Critical Points
Critical points are key locations on a function where its derivative is either zero or undefined. These points are crucial in the search for potential relative extrema.
  • When the derivative of a function is zero, this signifies a potential plateau or turning point on the graph, often leading to extrema.
  • If the derivative is undefined, it could indicate a corner or cusp, which also demands attention.
For the function given in the exercise, \( f(x) = 6x - x^2 \), the first derivative \( f'(x) = 6 - 2x \) equals zero at \( x = 3 \). Hence, \( x = 3 \) is identified as a critical point. These points are the starting block when analyzing and confirming extrema.
Power Rule
The Power Rule is a simple and powerful method for finding the derivative of polynomial functions. It states that if you have a term \( ax^n \), its derivative will be \( nax^{n-1} \).
  • In the case of \( f(x)=6x-x^2 \), applying the power rule to each term separately gives \( f'(x) = 6 - 2x \).
  • For the second derivative, applying the power rule again to \( f(x) = 6 - 2x \) results in \( f''(x) = -2 \).
  • This rule simplifies the process of differentiation, making it quick and straightforward to find derivatives critical for further analysis.
Understanding the power rule is crucial when working with derivatives as it applies to almost any polynomial function you encounter.
Concavity
Concavity describes the direction and curvature of a function鈥檚 graph and can be determined using the second derivative. Concavity indicates where a function is bending upwards or downwards, directly impacting the location of extrema.
  • A function is concave up if its second derivative is positive, suggesting a local minimum might be present.
  • Conversely, a function is concave down if the second derivative is negative, pointing towards a local maximum.
  • In the exercise, the second derivative \( f''(x) = -2 \) implies concave down across its domain, confirming \( x=3 \) as a relative maximum.
By analyzing the concavity, one determines not only the nature but also the confidence in identifying whether an extremum is indeed a maximum or minimum. It consolidates our understanding of the function's shape.

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Most popular questions from this chapter

True or False? In Exercises \(75-78\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.\(\begin{array}{l}{\text { The graph of every cubic polynomial has precisely one point }} \\ {\text { of inflection. }}\end{array}\)

Rolle's Theorem Let \(f\) be continuous on \([a, b]\) and differentiable on \((a, b) .\) Also, suppose that \(f(a)=f(b)\) and that \(c\) is a real number in the interval \((a, b)\) such that \(f^{\prime}(c)=0 .\) Find an interval for the function \(g\) over which Rolle's Theorem can be applied, and find the corresponding critical number of \(g\) , where \(k\) is a constant. (a) $$g(x)=f(x)+k$$ (b) $$g(x)=f(x-k)$$ (c) $$g(x)=f(k x)$$

Temperature When an object is removed from a furnace and placed in an environment with a constant temperature of \(90^{\circ} \mathrm{F},\) its core temperature is \(1500^{\circ} \mathrm{F} .\) Five hours later, the core temperature is \(390^{\circ} \mathrm{F}\) . Explain why there must exist a time in the interval \((0,5)\) when the temperature is decreasing at a rate of \(222^{\circ} \mathrm{F}\) per hour.

Think About It In Exercises \(79-82,\) create a function whose graph has the given characteristics. (There is more than one correct answer.) $$\begin{array}{l}{\text { Vertical asymptote: } x=2} \\ {\text { Slant asymptote: } y=-x}\end{array}$$

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