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Chapter 4: Problem 18

Find the exact global maximum and minimum values of the function. The domain is all real numbers unless otherwise specified. $$f(t)=\frac{t}{1+t^{2}}$$

Short Answer

Expert verified
The global maximum is \( \frac{1}{2} \) at \( t=1 \) and the global minimum is \( \frac{-1}{2} \) at \( t=-1 \).

Step by step solution

01

Find the derivative

The first step is to find the derivative of the function to determine the critical points. Given the function \( f(t) = \frac{t}{1+t^2} \), we will use the quotient rule for differentiation which states that if \( f(t) = \frac{u}{v} \), then \( f'(t) = \frac{v u' - u v'}{v^2} \). Here, \( u = t \) and \( v = 1 + t^2 \). Thus, the derivatives are \( u'=1 \) and \( v'=2t \). The derivative is:\[ f'(t) = \frac{(1 + t^2)(1) - t(2t)}{(1+t^2)^2} = \frac{1 + t^2 - 2t^2}{(1+t^2)^2} = \frac{1 - t^2}{(1+t^2)^2}. \]
02

Find critical points

To find the critical points, solve \( f'(t) = 0 \). Setting the numerator of the derivative equal to zero gives \( 1 - t^2 = 0 \). Solving for \( t \), we get the critical points:\[1 = t^2 \t = \pm 1.\]
03

Evaluate function at critical points and limits

Evaluate \( f(t) \) at the critical points and examine the behavior as \( t \to \infty \) and \( t \to -\infty \).- At \( t = 1 \): \( f(1) = \frac{1}{1+1^2} = \frac{1}{2} \).- At \( t = -1 \): \( f(-1) = \frac{-1}{1+(-1)^2} = \frac{-1}{2} \).As \( t \to \infty \) or \( t \to -\infty \), the term \( \frac{t}{1+t^2} \) approaches \( 0 \). Therefore, the function bounds itself at zero.
04

Determine global maximum and minimum

From the evaluations, the maximum value at the critical points is \( \frac{1}{2} \) and the minimum value is \( \frac{-1}{2} \). Since the function values as \( t \to \infty \) or \( t \to -\infty \) are \( 0 \), and \( \frac{-1}{2} < 0 < \frac{1}{2} \), these are the global extrema.

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Key Concepts

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

Derivative
To understand the global extrema of a function, the first step is often to find its derivative. The derivative of a function gives us important information about the rate at which the function changes. For a function like \( f(t) = \frac{t}{1+t^2} \), finding the derivative involves calculus techniques.
  • The derivative helps identify points where the function's slope is zero, these points are potential extrema (maximum or minimum).
  • For a quotient of two functions, like our example, the quotient rule helps in differentiation.
With the derivative, you can explore changes in the original function's graph and understand where it might reach its highest or lowest points. Knowing how to derive a function is fundamental in calculus for deeper insights into function behavior.
Critical Points
Critical points are the values of \( t \) where the derivative of the function equals zero or where the derivative does not exist. These points are significant because they are candidates for local and global extrema.
  • To find critical points for \( f(t) = \frac{t}{1+t^2} \), we locate where \( f'(t) = 0 \).
  • Setting \( 1 - t^2 = 0 \) gives us the critical points \( t = \pm 1 \).
Critical points tell us where to look on a graph for potential peaks or valleys. These are essential for determining where the global maximum and minimum of the function might occur.
Quotient Rule
The quotient rule is a technique for differentiating functions that are divided by each other. When you have a function \( \frac{u}{v} \), the rule provides a structured way to find the derivative. It is essential because it handles the complexities intrinsic to the ratios of functions.
  • It states: \( f'(t) = \frac{v u' - u v'}{v^2} \).
  • This formula is key in finding derivatives where both the numerator and denominator are functions of \( t \).
The quotient rule helps simplify the process, letting you manage more easily the sometimes complex algebra involved in differentiation. Understanding and mastering it is vital to tackling such problems.
Limits
Limits are crucial in defining the behavior of functions as a variable approaches a certain value or infinity. They help in understanding the end behavior of the function and are essential in determining global extrema by analyzing function behavior at infinity.
  • For \( f(t) = \frac{t}{1+t^2} \), investigating the limits as \( t \to \infty \) and \( t \to -\infty \) reveals how the function behaves at these extremes.
  • In this case, both limits approach 0, giving insight into the global bounds of the function.
By analyzing these limits, we decide whether the values at infinity impact the global extrema. This can confirm if the function reaches a maximum or minimum outside the critical points. Understanding limits helps in predicting the function鈥檚 behavior beyond immediate, observable range.

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Most popular questions from this chapter

Are the statements true of false? Give an explanation for your answer. If the radius of a circle is increasing at a constant rate, then so is the circumference.

Investigate the given two parameter family of functions. Assume that \(a\) and \(b\) are positive. (a) Graph \(f(x)\) using \(b=1\) and three different values for \(a\). (b) Graph \(f(x)\) using \(a=1\) and three different values for \(b\). (c) In the graphs in parts (a) and (b), how do the critical points of \(f\) appear to move as \(a\) increases? As \(b\) increases? (d) Find a formula for the \(x\) -coordinates of the critical point(s) of \(f\) in terms of \(a\) and \(b\). $$f(x)=(x-a)^{2}+b.$$

Are the statements in Problems true or false for a function \(f\) whose domain is all real numbers? If a statement is true,explain how you know. If a statement is false, give a counterexample. If \(x=p\) is not a local maximum of \(f,\) then \(x=p\) is not a critical point of \(f\).

An organism has size \(W\) at time \(t .\) For positive constants \(A, b,\) and \(c,\) the Gompertz growth function gives $$W=A e^{-e^{b-c t}}, \quad t \geq 0$$ (a) Find the intercepts and asymptotes. (b) Find the critical points and inflection points. (c) Graph \(W\) for various values of \(A, b,\) and \(c .\) (d) A certain organism grows fastest when it is about \(1 / 3\) of its final size. Would the Gompertz growth function be useful in modeling its growth? Explain.

Are the statements in Problems true or false for a function \(f\) whose domain is all real numbers? If a statement is true,explain how you know. If a statement is false, give a counterexample. If \(f^{\prime}(p)=0,\) then \(f(x)\) has a local minimum or local maximum at \(x=p\).
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